{ "schema": "projectforty2.erdos_frontier.v1", "source": { "name": "Erdős Frontier Atlas", "repository": "https://github.com/techno-optimist/erdos-frontier-atlas", "canonical_index": "https://www.erdosproblems.com", "doi": "10.5281/zenodo.21443635", "commit": "0394e3d", "generated_at": "2026-08-21T22:27:52Z", "note": "Official problem status belongs to erdosproblems.com. ProjectForty2 tracks bounded certificates, walls, and map state — not claimed prize solutions." }, "summary": { "hub_problems": 1217, "open": 619, "solved_upstream": 566, "walls": 27, "movable": 5, "cash_prize": 106, "in_lean": 404, "deep_audits": 51, "board_ready": 13, "board_heavy": 14, "bounded_quantities": 225, "graph_nodes": 2415, "graph_edges": 4007, "certificate_contracts": 15, "claimed_prize_solutions": 0, "board_receipts": 13 }, "board": [ { "tier": "certified", "problem": "#699 binomial-gcd rows", "problem_ids": [ 699 ], "result": "property holds for every n in [10⁷, 10⁸) — all 90,000,000 rows, every (i,j) pair decided exactly, zero counterexamples, 119.53 core-hours. First exhaustive general-row sweep past 10⁷ (structured families were already covered to ≈1.34×10⁸). Rests on a theorem that large primes never suffice on a pruned row — p=2 can be the only usable one", "certificate": "`certificates/erdos-699` · PR #129", "when": "2026-07-27" }, { "tier": "certified", "problem": "#366 2-full/3-full neighbours", "problem_ids": [ 366 ], "result": "no 2-full n with n+1 3-full for n ≤ 10²⁵ — 1,620,172,043 cubefull candidates, both orientations, 6.27 core-hours. A 1000× extension of 10²², which was a single unreplayed 2011 OEIS b-file; this re-derives that region independently. The lever: enumerate the cubefull side (~X^(1/3)), not the powerful-pair side (~X^(1/2)) that set the old bound", "certificate": "`certificates/erdos-366` · PR #125", "when": "2026-07-26" }, { "tier": "certified", "problem": "#743 Gyárfás tree packing", "problem_ids": [ 743 ], "result": "every one of the 45,376,056 tuples (T₂,…,T₁₀) decomposes K₁₀ — exhaustive, uncapped, 44 core-seconds. Frontier had stood at n ≤ 9 since Fishburn 1983 (43 years); n=9 reproduced as positive control. Hardest tuples are exactly those with T₄..T₇ all stars — where Gyárfás–Lehel's coverage stops", "certificate": "`certificates/erdos-743` · PR #126", "when": "2026-07-27" }, { "tier": "certified", "problem": "#993 tree independence unimodality", "problem_ids": [ 993 ], "result": "all 23,522,619,475 trees on n ≤ 30 unimodal, zero violations, 10.33 core-hours. Includes the first independent replication of the n ≤ 29 frontier (Reynolds, Zenodo v3, 8,691,747,673 trees — single-author, unreplayed until now), reaching his exact total. n=30 alone is 1.71× that entire prior workload", "certificate": "`certificates/erdos-993` · PR #126", "when": "2026-07-27" }, { "tier": "certified", "problem": "#552 R(C4,K1,n)", "problem_ids": [ 552 ], "result": "certified C₄-free witnesses ⇒ R(C4,K1,n) = n + ⌈√n⌉ + 1 for 12 ≤ n ≤ 16; n=17 closed at 22 (Parsons 1975)", "certificate": "`certificates/erdos-552` · PR #78", "when": "2026-07-16" }, { "tier": "certified", "problem": "#241 B₃-subset table (A387704)", "problem_ids": [ 241 ], "result": "proved A387704(n) = max{k : A227358(k) ≤ n−1} (translation invariance; 0/151 mismatches) ⇒ first jump to 9 at n=209; atlas cell closed by cross-reference", "certificate": "PR #80", "when": "2026-07-16" }, { "tier": "certified", "problem": "#13 Erdős–Sárközy", "problem_ids": [ 13 ], "result": "certified exact table f(1..45); N=17 is the last exception to ⌊N/3⌋+1 — an empirical location for Bedert's ineffective threshold", "certificate": "`certificates/erdos-13` · PR #81", "when": "2026-07-17" }, { "tier": "certified", "problem": "#979 f₃ / A385316", "problem_ids": [ 979 ], "result": "a(6) > 10¹² at C2 — exhaustively verified and replayable from this repo (verify.py --cutoff 1e12, ~80 s, ~11 GB; reproduces a(1..5) as a fail-closed self-check) — past the published 4.99·10¹¹. A stronger > 10¹³ sweep exists but its code and ledgers are not tracked here, so it is quarantined from public promotion pending a self-contained replay packet — see the entry note in the gap map. (Corrected 2026-07-25: this row previously claimed > 10¹³ at C1; the second \"independent implementation\" backing that class lives outside this repository, so a reader could not replay it.)", "certificate": "`certificates/erdos-979`", "when": "2026-07-19" }, { "tier": "grounded", "problem": "#1107 Mollin–Walsh / A056828", "problem_ids": [ 1107 ], "result": "verified no seventh exception below 10⁶ to being a sum of ≤3 powerful numbers (verify.py, default N = 10⁶, ~10 s, dependency-free); the six known exceptions {7,15,23,87,111,119} are all < 120 and are reproduced, powerful-counts cross-checked vs A118896. A wider 10¹⁰ run exists but is not replayable from this repository, so it is not the public claim. (Corrected 2026-07-25: this row previously claimed 10¹⁰.)", "certificate": "`certificates/erdos-1107`", "when": "2026-07-18" }, { "tier": "grounded", "problem": "#142 r₃(N)", "problem_ids": [ 142 ], "result": "complete 12,349-cell geometric enumeration superseding a flawed 976-cell subset, now certified in-repo as a construction no-go — a foundation only; self-declared no-bridge, not an r₃(N) bound", "certificate": "`certificates/erdos-142` · PR #84", "when": "2026-07-13" }, { "tier": "corrected", "problem": "#142 / D15 lemmas", "problem_ids": [ 142 ], "result": "refuted ker π ∩ D = 0 and q ≥ dim ker π (two lemmas a bridge attempt rested on) and proved Theorem A in their place — a dead path closed so the next agent does not re-walk it; explicitly not an r₃(N) bound", "certificate": "`certificates/erdos-142-kerpi-refutation` · PR #101, #102", "when": "2026-07-24" }, { "tier": "certified", "problem": "#1029 / #77 R(5,5)", "problem_ids": [ 1029, 77 ], "result": "42/42 DRAT-certified structural negatives (no witness; rigidity + prime-order orbit collapse), all consistent with R(5,5) = 43", "certificate": "r55-rigidity-certificates · DOI 10.5281/zenodo.21305022", "when": "2026-07-10" }, { "tier": "corrected", "problem": "#552 R(C4,K1,39)", "problem_ids": [ 552 ], "result": "the =46 new-value claim was retracted — DS1 rev.18 lists 46 ≤ f(39) ≤ 47, OPEN; the 45-vertex witness stands as a re-derivation of Wu–Sun–Radziszowski 2015", "certificate": "`certificates/erdos-552-f39`", "when": "2026-07-17" } ], "deep_audits": [ { "id": 1, "title": "Erdős distinct subset sums (min largest element, A276661)", "prize": "$500 (for the asymptotic headline conjecture only; the finite a(10) determination carries no prize)", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = 11 integers; 2048 subset sums, exact and fast; live bracket 462 <= a(11) <= 594", "frontier": "462 <= a(11) <= 594", "verifier": "Given a candidate 11-set, enumerate all 2048 subset sums and test pairwise distinctness with exact integers. Witness verification is deterministic and fast; optimality remains the expensive search claim.", "finding": "", "erdos_url": "https://www.erdosproblems.com/1", "oeis": [ "A005318", "A276661" ], "arxiv": [ "2208.12182", "2308.03748" ] }, { "id": 2, "title": "Covering-system min-modulus record (>42)", "prize": "$1000", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "witness not representable: a min-modulus-40+ covering system already has > 10^50 recursively-defined congruences; no flat poly-time check", "frontier": "", "verifier": "Checking a covering system covers Z = verifying it covers all residues mod L (L = lcm of moduli). But L is a product of primes far beyond enumeration (astronomically large), so verification is NOT clean poly-time residue-enumeration — it requires a structured CRT/inclusion-exclusion/recursion argument. Cost: intractable to enumerate; only structured symbolic verification works, and only because the object is recursively defined.", "finding": "", "erdos_url": "https://www.erdosproblems.com/2", "oeis": [], "arxiv": [] }, { "id": 13, "title": "Erdős–Sárközy no-term-divides-sum sets (|A|≤N/3+O(1))", "prize": "$100 (already awarded to Bedert)", "beatable": "MOVABLE", "board_class": "HEAVY", "board_class_reason": "no defended record to beat (uncharted table); the boardable product is a certified exact-value table - ILP/SAT certificate claims, not single witnesses (R2 fails)", "frontier": "Bedert (arXiv:2301.07065) proved |A|<=floor(N/3)+1 for large N (WALL, ineffective threshold). CERTIFIED small-N table (certificates/erdos-13/, 2026-07-17): f(1..45) exact, two independent methods; f(N) beats floor(N/3)+1 exactly at N=2,4,5,7,8,10,11,13,14,17 and equals it for 18<=N<=45 (empirical threshold; N=17 the last exception, witness {6,8,9,11,12,14,17}). Not in OEIS; candidate sequence PENDING full literature/definition novelty-check before external submission.", "verifier": "Given A, check all triples a= 55 (1972), Sun(3,4) >= 39 (1992), Sun(3,5) >= 89 (1974) - construction records static 30-50 yr", "verifier": "Given N sets: certify no s of them form a sunflower. For s=3, enumerate triples grouped by common core — O(N^3 · m) worst case, far less with core-bucketing. Exact, poly-time for fixed s, cheap (seconds for N in the hundreds).", "finding": "", "erdos_url": "https://www.erdosproblems.com/20", "oeis": [ "A332077" ], "arxiv": [] }, { "id": 21, "title": "q(6): min edges, 6-uniform intersecting hypergraph with τ=6", "prize": "$500 — but already PAID to Kahn (1994) for the asymptotic linear upper bound. Determining q(6) claims NO prize; it is a real but non-prize combinatorial contribution.", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = <= 64 six-sets; pairwise-intersect + complete 5-cover branch-and-bound <= 9,331 nodes, ms-exact; open bracket 14..18 witness-improvable (m <= 17 wins)", "frontier": "14 <= q(6) <= 18", "verifier": "Poly-time exact. Given edge list (m 6-sets): (a) pairwise-intersecting — O(m²·6); (b) τ=6 i.e. no 5-cover — a fixed-size-5 hitting-set check over ≤N vertices (N≲6m), either C(N,5) enumeration or a tiny ILP; unsatisfiable ⇒ τ≥6, and τ≤6 is automatic since any edge covers. Cost: milliseconds per candidate.", "finding": "2026-07-12: we attempted q(6) with our SAT+DRAT pipeline (the R(5,5) method). It FAILED: q(r) is tau-critical, valid objects are vanishingly rare among m-edge intersecting families, and the tau>=r encoding is a C(V,r-1) blowup whose resolution proofs explode. D(5,12) (proving the KNOWN q(5)>=13) timed out at 3600s/6.4GB; D(6,14) timed out at every proven vertex count V in [21,29] (up to 8.4M clauses) — kissat could not even decide it. R(5,5) worked because its constraints are LOCAL (SAT-friendly); q(r)'s are GLOBAL/rare (SAT-hostile). The correct tool is ORDERLY GENERATION (canonical augmentation + tau-deficiency prune), which is what Barat used. The P42 board STAYS (the exact verifier works fine) as a MARKET-MAKER board: the frontier is movable, but by generation or cluster cube-and-conquer, not by our SAT machinery. Vertex bounds for the generation attack: Barat Lemma 6.4 (e=14 => V<=29), Obs 6.1/6.2 degree caps; Sivashankar q(6)>=14 means a single 14-edge witness would give q(6)=14 exactly.", "erdos_url": "https://www.erdosproblems.com/21", "oeis": [], "arxiv": [ "2011.04444", "2503.14918", "2606.24878" ] }, { "id": 27, "title": "ε-almost covering systems with moduli in [N, CN]", "prize": "$100", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "headline DISPROVED (Filaseta-Ford-Konyagin-Pomerance-Yu 2007); extremal witnesses not finitely representable", "frontier": "", "verifier": "Checking that a proposed system covers Z (or leaves density ≤ ε). But the natural extremal object has >10^50 moduli and an astronomically large LCM period, so there is no flat enumerable poly-time checker over a period; verification requires the recursive distributed-covering structure of the construction itself. This absence of a clean small-object exact verifier is the disqualifier for our verifier-first farm.", "finding": "", "erdos_url": "https://www.erdosproblems.com/27", "oeis": [], "arxiv": [] }, { "id": 30, "title": "Erdős #30 — max Sidon set h(N)", "prize": "$1000", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "frontier is proven-optimal (OGR-28); the next move is an optimality/exhaustion claim adjudicated by receipts, not a witness", "frontier": "OGR-28 (length 585) proven optimal Nov 2022 by distributed.net after 8.5 yr; next move (OGR-29) is an exhaustion claim", "verifier": "(A) Construction check: enumerate all C(k,2) pairwise differences of a claimed size-k Sidon set, confirm distinctness — O(k^2), trivial. But the RECORD is the proven-optimal value, whose nonexistence side has no compact certificate (distributed.net emits a brute-force search log, not a DRAT-style checkable proof). (B) Upper-bound constant: verify a finite rational certificate (dual feasibility of the kernel QP) in exact arithmetic — cheap, seconds.", "finding": "", "erdos_url": "https://www.erdosproblems.com/30", "oeis": [], "arxiv": [ "2607.01169" ] }, { "id": 39, "title": "Infinite near-maximal-density Sidon set", "prize": "$500 (Erdős)", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "optimality claims (branch-and-bound nonexistence receipts) are the movable half; witness side has no open tracked value", "frontier": "Sidon/Golomb optimality beyond OGR-28: exhaustion-certificate tier; asymptotic headline (Ruzsa exponent ~0.4142) untouchable", "verifier": "Sidon/Golomb-ruler check: all C(k,2) pairwise differences distinct — O(k^2) exact integer arithmetic. Trivially poly. Optimality of a size/length claim requires exhaustive nonexistence over all shorter rulers (branch-and-bound), which is the expensive part.", "finding": "", "erdos_url": "https://www.erdosproblems.com/39", "oeis": [ "A004137" ], "arxiv": [] }, { "id": 41, "title": "Distinct triple-sum (B_3) sets — infinite density liminf", "prize": "$500 (headline only; unreachable by finite computation — asymptotic theorem required)", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = 11 integers; 286 triple-sum distinctness checks, exact, microseconds; open frontier 445 witness-improvable", "frontier": "a(11) <= 445 (Tromp 2013, A227358); optimality open, suspected < 440", "verifier": "Enumerate all C(n+2,3)=286 triple sums (multisets) for the 11 marks and check pairwise distinctness — O(n^3), ~10^2–10^3 ops, negligible. Optimality (no 11-mark B_3 set in [0,L*-1]) is encoded as SAT and certified with a DRAT proof, exactly as in our R(5,5) nonexistence work.", "finding": "", "erdos_url": "https://www.erdosproblems.com/41", "oeis": [ "A227358" ], "arxiv": [ "2308.12406" ] }, { "id": 43, "title": "Disjoint-difference Sidon set pairs", "prize": "$100 (already effectively claimed via disproof)", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "problem DISPROVED (both questions resolved negatively); no movable number", "frontier": "", "verifier": "Poly-time exact: verify each set is Sidon (all C(n,2) pairwise differences distinct, O(n^2)); verify (A-A)∩(B-B)={0} by hashing difference multisets and intersecting, O(n^2). Cheap. But no target record to certify against.", "finding": "", "erdos_url": "https://www.erdosproblems.com/43", "oeis": [], "arxiv": [] }, { "id": 52, "title": "Erdős–Szemerédi sum-product", "prize": "$250", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "no concrete n, authoritative seed witness, or tracked per-n record is identified; the asymptotic headline is not advanced by an unscoped finite score (R2 fails)", "frontier": "A263996 small-case table (smallest |A+A u AA| over n-sets); contest-set records", "verifier": "Trivial exact checker: given A, compute the multiset of pairwise sums and products, dedupe, take union cardinality. O(n^2) integer arithmetic (bignum since products blow up), microseconds even for n in the thousands. Perfect poly-time exact verifier — the strongest part of the fit.", "finding": "", "erdos_url": "https://www.erdosproblems.com/52", "oeis": [ "A263996" ], "arxiv": [] }, { "id": 64, "title": "Erdős–Gyárfás 2-power cycle conjecture", "prize": "$1000 (erdosproblems.com listing; Erdős originally $100 to prove / $50 to disprove)", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "movable claim = exhaustive-generation certificate ('no cubic counterexample <= 30 vertices'); a counterexample itself would be witness-cheap but is believed nonexistent (existence bounty, no frontier)", "frontier": "no counterexample: verified for all cubic graphs <= 28 vertices (Royle-Markstrom ~2004); extension = exhaustive-generation certificate", "verifier": "For a fixed graph, detect existence of a cycle of each length 2^k <= n. In bounded-degree (cubic) graphs, C4/C8 detection is trivial and C16/C32 detection is poly-time (color-coding / bounded DP). Cost is negligible per graph; almost all graphs are eliminated by the cheap C4 test, so amortized cost is tiny. No DRAT needed — nonexistence is an exhaustive-generation certificate (genreg/snarkhunter enumeration is the proof).", "finding": "", "erdos_url": "https://www.erdosproblems.com/64", "oeis": [], "arxiv": [] }, { "id": 67, "title": "Erdős Discrepancy Problem", "prize": "$500 — but ALREADY CLAIMED by Tao (problem PROVED). No prize is obtainable by any finite computation.", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = +-1 sequence; O(N log N) integer prefix-sum scan; exactly re-verified 130,000-term frontier witness-extendable", "frontier": "C=3 general witness >= 130,000 (Konev-Lisitsa unrestricted witness, exactly re-verified by P42); C=2 tight at 1160", "verifier": "Exact and near-free: given a candidate sequence of length n, verify max over all d≤n of running prefix-sums along the d-subsampled sequence stays ≤C. O(n log n) integer arithmetic; DRAT for any UNSAT (nonexistence) step. Canonical verifier-first fit.", "finding": "2026-07-12 correction: the unrestricted Konev-Lisitsa length-130,000 witness was recovered from the authors' archived result artifact and exactly re-verified by the P42 verifier. Earlier ~14,000 frontier and search-cost prose in the originating audit is superseded; new work must start above 130,000.", "erdos_url": "https://www.erdosproblems.com/67", "oeis": [], "arxiv": [] }, { "id": 77, "title": "lim R(k)^{1/k} (diagonal Ramsey growth constant)", "prize": "$250", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "asymptotic limit only; the finite R(5,5) frontier object is carried by #1029 (R4 dedupe)", "frontier": "", "verifier": "O(n^k) / clique-check on a fixed graph: verifying a candidate Ramsey graph of order n has no monochromatic K_k is polynomial in n for fixed k (enumerate k-subsets). Exact and cheap per-order; this is exactly our R(5,5) verifier. But it certifies one small value, not the limit.", "finding": "", "erdos_url": "https://www.erdosproblems.com/77", "oeis": [], "arxiv": [] }, { "id": 83, "title": "EKR t=2 intersecting family bound on [4n]", "prize": "$500 (already awarded/solved)", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "proved theorem (Ahlswede-Khachatrian 1997); a MIS solve certifies nothing new", "frontier": "", "verifier": "Poly-time count-and-check of a candidate family (verify sizes, pairwise intersections ≥ 2, count). Exact maximality would need max-independent-set on the conflict graph (vertices = 2n-subsets, edges = intersection ≤ 1). Cost blows up: n=3 → 924 vertices, n=4 → 12870 vertices; and the answer is already a proved theorem, so the MIS solve certifies nothing new.", "finding": "", "erdos_url": "https://www.erdosproblems.com/83", "oeis": [ "A071799" ], "arxiv": [] }, { "id": 86, "title": "C4-free subgraphs of the hypercube (Turán density 1/2?)", "prize": "$100 (attaches ONLY to the asymptotic conjecture, which is NONE — unclaimable by finite computation)", "beatable": "UNKNOWN", "board_class": "READY", "board_class_reason": "witness = edge subset of Q7; scan the 672 4-cycles, exact, microseconds; ex(Q7,C4) >= 305 witness-improvable", "frontier": "ex(Q7,C4) >= 304, ex(Q8,C4) >= 680 (SA, arXiv:2603.29127, May 2026); exact values open past ex(Q6,C4)=132", "verifier": "C_4-freeness check: enumerate the fixed 4-cycles of Q_n (672 in Q7, 1792 in Q8) and verify none is fully contained; O(#cycles) per candidate, microseconds. Trivially exact and cheap.", "finding": "", "erdos_url": "https://www.erdosproblems.com/86", "oeis": [ "A245762" ], "arxiv": [ "2603.29127" ] }, { "id": 90, "title": "Unit distance problem — max unit pairs (A186705)", "prize": "$500 (headline) — already effectively resolved: DISPROVED in the negative, Lean-verified, by an OpenAI internal model. Earlier $300/$250 sub-offers were for the asymptotic n^{1+o(1)} bound, now settled. No prize attaches to the finite OEIS-extension sub-problem.", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "exact-value verifier is ExistsR-hard (unit-distance realizability); witness side requires algebraic coordinates - equality constraint makes the rational restriction lossy (R3 fails)", "frontier": "", "verifier": "Two directions, asymmetric cost. LOWER bound: given explicit (rational/algebraic) coordinates, count pairs at distance exactly 1 — poly-time O(n^2), clean fit. EXACTNESS/upper: decide whether an abstract candidate graph is realizable as a unit-distance graph in R^2 — this is ∃R-hard (existential theory of the reals), NOT poly-time. The specialists use a custom, incomplete embeddability solver that returns 'I don't know' for girth-≥5 non-UDGs. This realizability oracle is the true cost and the wall.", "finding": "", "erdos_url": "https://www.erdosproblems.com/90", "oeis": [ "A186705" ], "arxiv": [ "2406.15317", "2412.11914" ] }, { "id": 92, "title": "Max-equidistance-count planar point sets (Erdős–Fishburn table)", "prize": "$500 (disproved upper bound) / $50 (counterexample, per ErFi97) — both effectively resolved/dead", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "headline disproved with #90; exact-value direction has no finite verifier; equality-constrained coordinates (R3 fails)", "frontier": "", "verifier": "Construction side only: given exact rational/algebraic coordinates of n points, compute all C(n,2) squared distances and check each point has some distance value with multiplicity >= k. O(n^2) exact, trivial. This verifies UPPER bounds on g(k) (constructions). NO finite verifier exists for the nonexistence/exact-value direction.", "finding": "", "erdos_url": "https://www.erdosproblems.com/92", "oeis": [], "arxiv": [] }, { "id": 97, "title": "Convex polygon: every vertex has 4 equidistant others", "prize": "$100", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "existence bounty with no numeric frontier, and the defining condition (4 equidistant points) is equality-constrained real geometry (R3 fails: rational restriction lossy)", "frontier": "", "verifier": "Given rational vertex coords: (a) verify convex position (O(n) orientation checks) and (b) for each vertex, group squared distances to others and check some value has multiplicity >=4 (sort per vertex). Exact, O(n^2 log n). Cheap and clean.", "finding": "", "erdos_url": "https://www.erdosproblems.com/97", "oeis": [], "arxiv": [ "2511.02864" ] }, { "id": 101, "title": "Erdős #101 — 4-point lines o(n²), orchard A006065", "prize": "$100", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "collinearity is an equality constraint: optimal configurations may require irrational coordinates (ExistsR realizability gap), so a rational-witness board is not lossless (R3 fails); nonexistence side has no DRAT route", "frontier": "", "verifier": "Exact rational determinant collinearity checker: for a candidate n-point set compute all C(n,4) collinearity tests (or sort slopes per point), count size-exactly-4 lines, assert no line has ≥5. O(n^4), microseconds for n≤30. Poly-time, cheap.", "finding": "", "erdos_url": "https://www.erdosproblems.com/101", "oeis": [ "A006065", "A172992" ], "arxiv": [] }, { "id": 107, "title": "Erdős–Szekeres \"Happy Ending\" convex-polygon conjecture", "prize": "$500", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "orientation checking is exact, but no a-priori coordinate bit bound is established; a byte-capped integer format could exclude valid order types (R1/R3 admission proof pending)", "frontier": "33 <= f(7) <= 127 (a 33-point general-position set with no convex 7-gon would refute Erdos-Szekeres at n=7)", "verifier": "Convexity check: a k-set is in convex position iff all its 4-subsets are convex (poly-time). Checking a 33-set contains a convex 7-gon = scan C(33,7)≈4.3M 7-subsets (or via the order type) — cheap, exact, DRAT-certifiable in the SAT encoding. Excellent verifier; this is exactly why the form maps to our farm.", "finding": "", "erdos_url": "https://www.erdosproblems.com/107", "oeis": [], "arxiv": [ "2512.24061" ] }, { "id": 114, "title": "Erdős–Herzog–Piranian lemniscate max length", "prize": "$250 (Erdős); falsification-only prize (disprove by counterexample).", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "transcendental objective (elliptic-integral arc length); no exact/DRAT verifier exists; Tao proved the conjecture for large n (Dec 2025)", "frontier": "", "verifier": "Compute arc length of the level set {|p(z)|=1}: trace the (possibly multi-oval) curve and integrate ds numerically. NOT poly-time-exact — it is transcendental; rigorous certification would need interval-arithmetic enclosures of an elliptic-type integral over the whole level set. Cost: cheap for a heuristic float estimate, expensive and non-standard for a certified enclosure; no DRAT analog exists.", "finding": "", "erdos_url": "https://www.erdosproblems.com/114", "oeis": [], "arxiv": [ "2512.12455" ] }, { "id": 128, "title": "Erdős Sparse Half Conjecture (triangle-free)", "prize": "$250", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "verifier is NP-hard (Sparsest-k-Subgraph); meaningful regime is flag-algebra SDP blowups, not finite witnesses", "frontier": "", "verifier": "Exact sparsest-half check = minimize induced edge count over all ⌊n/2⌋-subsets (NP-hard Sparsest-k-Subgraph). Cost: ILP/SAT feasible only for n≲40–60 per candidate; the meaningful regime is asymptotic blowups where the \"verifier\" becomes a fractional quadratic minimization over vertex weights (flag-algebra SDP).", "finding": "", "erdos_url": "https://www.erdosproblems.com/128", "oeis": [], "arxiv": [] }, { "id": 138, "title": "van der Waerden numbers W(k)", "prize": "$500 (for improving bounds / proving W(k)^{1/k} -> infinity)", "beatable": "WALL", "board_class": "READY", "board_class_reason": "witness = 2-coloring of [N]; O(N^2/k) mono-AP scan, exact; W(2,7) > 3703 witness-improvable", "frontier": "W(2,7) >= 3703 (Ahmed et al. 2014); W(2,6)=1132 exact (Kouril-Paul 2008)", "verifier": "Poly-time exact: scan a candidate coloring for a monochromatic k-AP in O(N^2/k). Trivial and cheap - but it is a commodity every prior record-holder already runs, so no moat.", "finding": "", "erdos_url": "https://www.erdosproblems.com/138", "oeis": [], "arxiv": [] }, { "id": 139, "title": "Exact r_k(N): largest AP-free subset table (Erdős–Turán / Szemerédi)", "prize": "no", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "movable content is the exact-table certificate side (exhaustion receipts for r_k, k >= 4); the r_3(212) witness object is carried by #140", "frontier": "exact r_k(N) tables beyond r_3: certificate-tier (exhaustion receipts); r_3 frontier object carried by #140", "verifier": "Poly-time exact AP-freeness check: for candidate S⊆[1,N], confirm no a= 37 witness-improvable", "frontier": "R(4,6) in [36,40] (lower 36 = Exoo 2012)", "verifier": "Exact and cheap: given a candidate graph on n≤~80 vertices, check no K₄ (O(n⁴) clique enumeration) and no independent set of size k (bounded clique search on complement; trivial at these n). Nonexistence at a size via SAT + DRAT certificate over the C(n,2) edge variables with clique/independent-set clauses — the same encoding used for R(5,5). Verifier cost negligible; the search is the cost.", "finding": "", "erdos_url": "https://www.erdosproblems.com/166", "oeis": [], "arxiv": [ "2509.03784" ] }, { "id": 182, "title": "Max edges without a k-regular subgraph (Erdős–Sauer)", "prize": "$100 (offered by Erdős for k=3) — effectively resolved; not collectible via finite computation.", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "'contains a k-regular subgraph' is NP-complete for k >= 3: no poly-time exact verifier", "frontier": "", "verifier": "\"Contains a k-regular subgraph?\" is NP-COMPLETE for every k≥3 (Chvátal/Garey–Johnson; cubic-subgraph NP-c even on bipartite max-deg-4). So there is NO poly-time exact verifier. Existence of a k-regular subgraph has a trivial poly certificate (exhibit it); NON-existence needs a SAT-UNSAT/DRAT proof. Determining f_k(n) additionally requires the ∀-over-all-graphs upper bound (a second, harder exhaustion). Cost: per-graph SAT is cheap for n≲12 but the outer max-edge + all-graph search explodes.", "finding": "", "erdos_url": "https://www.erdosproblems.com/182", "oeis": [], "arxiv": [ "2411.11785" ] }, { "id": 183, "title": "Limit of multicolor triangle Ramsey R(3;k)^{1/k}", "prize": "$250 (Erdős)", "beatable": "WALL", "board_class": "READY", "board_class_reason": "witness = k-edge-coloring; per-color triangle scan O(k n^3), exact; R_4(3) >= 52 witness-improvable", "frontier": "R_4(3) in [51,62] (Chung 1973 lower), R_5(3) in [162,307] (Exoo)", "verifier": "Per-color triangle-freeness of an n-vertex k-edge-coloring: O(k·n^3) exact (or O(k·n^ω)). Nonexistence side = SAT/DRAT over the coloring CNF, infeasible at these n.", "finding": "", "erdos_url": "https://www.erdosproblems.com/183", "oeis": [ "A003323" ], "arxiv": [] }, { "id": 211, "title": "Erdős #211 — Beck/Szemerédi-Trotter (finite residue: orchard 3-point lines)", "prize": "$100 (Erdős #211, but problem is SOLVED — not claimable). Orchard sub-problem carries no formal prize.", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "same ExistsR/collinearity obstruction as #101 - orchard-type equality-constrained geometry (R3 fails)", "frontier": "", "verifier": "Exact and cheap: for a candidate point set, test every one of the C(n,3) triples for collinearity via an integer/rational 3×3 determinant, reject if any line holds ≥4 points, count exactly-3-point lines. O(n³) = 455 determinant checks at n=15 — trivial, fully exact. (Verifier is clean; it's the SEARCH/nonexistence-certificate that is the wall.)", "finding": "", "erdos_url": "https://www.erdosproblems.com/211", "oeis": [ "A003035" ], "arxiv": [ "1208.4714" ] }, { "id": 241, "title": "B_3 sets: max A⊆{1..N} with all triple sums a+b+c distinct; is f(N)~N^{1/3}?", "prize": "$100 (for a construction achieving ω(N)·N^{1/3} with ω→∞) plus a separate historical $25 (for N^c, c>1/3). Both are ASYMPTOTIC-construction prizes — a finite witness cannot claim either.", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = subset of {1..N}; O(k^3) triple-sum distinctness, exact; first-jump frontier witness-improvable", "frontier": "CLOSED BY CROSS-REFERENCE (2026-07-16): A387704(n) = max{k : A227358(k) <= n-1} (B_3 is translation-invariant, so a k-element B_3 subset of {1..n} exists iff a k-mark B_3 ruler of length <= n-1 exists). Identity machine-checked against all 151 published terms (0 mismatches). Hence the table is determined through n=309 by Tromp's exhaustive A227358 values: a(n)=8 for 130<=n<=208, FIRST JUMP TO 9 AT n=209, a(n)=9 for 209<=n<=309, jump to 10 at n=310. The only open content is the jump to 11 at A227358(11)+1 <= 446 — exactly problem #41's open cell a(11).", "verifier": "Given a candidate set A of size k: enumerate all C(k+2,3)=(k+2)(k+1)k/6 multiset triple-sums into a bitset over [3,3n] and check all distinct — O(k^3) exact, trivially cheap. Certifying f(n)=v (nonexistence of a size-(v+1) set in {1..n}) requires an EXHAUSTIVE branch-and-bound tree (DRAT-style search certificate), the expensive half.", "finding": "2026-07-16 CLOSED-BY-CROSSREF: A387704(n)=max{k:A227358(k)<=n-1} (translation invariance), machine-checked 0/151 mismatches; first jump to 9 = 209; table free to n=309; open content = #41 a(11). The 'hobbyist backtracker' table was A227358 in disguise — third stale-frontier catch of the day (#552 a(17), f(39), now this).", "erdos_url": "https://www.erdosproblems.com/241", "oeis": [ "A387704" ], "arxiv": [] }, { "id": 500, "title": "Turán tetrahedron K_4^3 extremal number", "prize": "$500 (Erdős — for the asymptotic density of any single K_ℓ^r, ℓ>r≥3; unclaimed)", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "finite lower bound is a closed-form construction (A140462) believed exact, not an open tracked frontier; density side carried by #712", "frontier": "", "verifier": "K_4^3-free check: for every one of the C(n,4) 4-subsets, at least one of its 4 triples is absent. O(n^4), exact, poly-time — an excellent cheap verifier. (Cost is trivial; it's the search/UNSAT-proof side that walls out.)", "finding": "", "erdos_url": "https://www.erdosproblems.com/500", "oeis": [ "A140462" ], "arxiv": [] }, { "id": 548, "title": "Erdős–Sós conjecture (trees in dense graphs)", "prize": "$100", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "no finite search space (n unbounded) and no numeric record; positive direction has no finite certificate", "frontier": "", "verifier": "Exact and cheap for a CLAIMED counterexample: edge-count check (trivial) + subgraph-isomorphism T↪G for a fixed (k+1)-vertex tree T, poly in n via color-coding / bounded-pattern search (O(n^{O(k)}) naive, FPT in k). The verifier is fine; the problem is there is no bounded search space to run it over (n unbounded) and the positive direction has no finite certificate.", "finding": "", "erdos_url": "https://www.erdosproblems.com/548", "oeis": [], "arxiv": [ "2405.15733" ] }, { "id": 552, "title": "R(C_4, S_n): C4-vs-star Ramsey", "prize": "$100 (Erdős)", "beatable": "MOVABLE", "board_class": "READY", "board_class_reason": "witness = C4-free red graph with minimum degree >= m-n; codegree and degree checks are exact and cheap; the next open cell a(17) is witness-improvable from the certified m=21 seed", "frontier": "Repository-certified: a(12..16) = 17,18,19,20,21. a(17) = 22 CLOSED (Parsons 1975: R(C4,K1,q^2+1)=q^2+q+2 at q=4; independently, the certified 21-vertex witness gives a(17)>=22 and min-degree-5 on 22 vertices needs 55 edges > ex(22;C4)=52 [OEIS A006855], so a(17)<=22). CORRECTION 2026-07-17: the earlier 'a(39)=46 NEW VALUE' claim is RETRACTED — DS1 rev.18 (Apr 2026) Table IVa lists a(39)='46-47' OPEN (lower 46 = Wu-Sun-Radziszowski 2015 Cons.5; upper 47 = Dybizbanski-Dzido); Boza arXiv:2409.12770's '/46' cell is an off-by-one wheel-convention error vs its own source. Our 45-vertex witness (certificates/erdos-552-f39/) re-derives the 2015 lower bound. Open deciders: a(39) via SAT cell (46-vtx min-deg-7: UNSAT=>46, SAT=>47); a(42) via (49,7); a(44) via (51,7). Consult DS1, not single-survey tables.", "verifier": "Represent the red graph. Check C_4-freeness by requiring every vertex pair to have codegree <= 1. A blue S_n is absent exactly when every blue degree is <= n-1, equivalently every red degree is >= m-n on an m-vertex complete graph. Both checks are exact and tiny for m ~ 15-45.", "finding": "2026-07-13: local PySAT/CaDiCaL edge-SAT passes found exact witnesses meeting Parsons' upper bound for n=12..16 and a 21-vertex witness proving R(C4,K1,17) >= 22. Dependency-free verifiers recomputed all degrees and pair-codegrees; CHRONOS independently replayed n=12 and a separate hostile checker replayed all six graphs. A limited 1,000,000-conflict pass on n=17,m=22 returned UNKNOWN, not UNSAT. Do not repeat the certified regions or treat that timeout as a bound. 2026-07-16 SOURCE-FRESHNESS CORRECTION: the '22<=a(17)<=23 open' framing was stale — f(17)=22 since Parsons 1975 (confirmed by Boza 2026 table and by counting against ex(22;C4)=52). The 1,000,000-conflict UNKNOWN on n=17,m=22 was a search for a witness that provably cannot exist. Frontier tracking must consult the survey literature (Radziszowski DS1, Boza) not only the OEIS b-file. 2026-07-17 RETRACTION: the f(39)=46 equality claim was caught by our own novelty gate BEFORE external submission — lower bound 46 was WSR 2015; Boza's upper '/46' is a convention error; DS1.18 says open in {46,47}. Witness stands as a re-derivation certificate. LESSON: verify against DS1 (the maintained survey), and never trust a single survey table for an equality claim.", "erdos_url": "https://www.erdosproblems.com/552", "oeis": [ "A006672" ], "arxiv": [] }, { "id": 564, "title": "Doubly-exp lower bound for 2-color 3-uniform Ramsey R_3(n)", "prize": "no", "beatable": "WALL", "board_class": "READY", "board_class_reason": "witness = 2-coloring of triples on 35+ vertices; O(m^5) scan of 4-/5-subsets, exact, cheap; R_3(4,5;3) >= 36 witness-improvable", "frontier": "R_3(4,5;3) >= 35 (SAT class-decomposition record)", "verifier": "Poly-time exact: given a coloring of the C(m,3) triples, scan all C(m,4) 4-subsets (no red K_4 = not all 4 triples red) and all C(m,5) 5-subsets (no blue K_5). O(m^5) checks, trivial. Nonexistence certifiable via SAT UNSAT + DRAT, exactly as R(5,5).", "finding": "", "erdos_url": "https://www.erdosproblems.com/564", "oeis": [], "arxiv": [] }, { "id": 582, "title": "Folkman number Fe(3,3;4)", "prize": "$100 (Erdős, for the original existence — already PROVED-in-Lean); separately Graham's $100 for proving/disproving N ≤ 100.", "beatable": "UNKNOWN", "board_class": "HEAVY", "board_class_reason": "witness (K4-free) is cheap, but adjudicating 'G arrows (3,3)' costs one large CDCL UNSAT + DRAT check per candidate - optimistic-oracle tier (R1 fails)", "frontier": "21 <= Fe(3,3;4) <= 786 (Lange-Radziszowski-Xu 2012); Graham's $100 for a K4-free graph <= 100 vertices arrowing (3,3)", "verifier": "K4-free: trivial poly check (enumerate 4-subsets or clique-free test). Arrowing: encode \"there EXISTS a 2-edge-coloring with no monochromatic triangle\" as SAT (one Boolean per edge; for each triangle two clauses forbidding all-red and all-blue). G arrows iff that formula is UNSAT. UNSAT is DRAT-certifiable. Cost: one large CDCL UNSAT per candidate — comparable scale to the R(5,5) window-UNSATs we already ran; hours-to-days for a ~200-800-vertex graph, DRAT check linear in proof size.", "finding": "", "erdos_url": "https://www.erdosproblems.com/582", "oeis": [], "arxiv": [] }, { "id": 588, "title": "Erdős #588 — f_k(n)=o(n²) for k≥4 (orchard-type k-rich lines)", "prize": "$100 (Erdős, [Er84])", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "same ExistsR/collinearity obstruction as #101 (R3 fails); the continuous search is unconstrained by the verifier", "frontier": "", "verifier": "Given n points with exact rational/algebraic coordinates: check every pair-generated line hits ≤k points (no k+1 collinear) and count lines with ≥k points. O(n²log n)–O(n³), trivially cheap. Poly-time exact — fits verifier-first, but only verifies; it does not constrain the continuous search.", "finding": "", "erdos_url": "https://www.erdosproblems.com/588", "oeis": [ "A006065", "A008997" ], "arxiv": [] }, { "id": 687, "title": "Erdős #687 — estimates for Y(x), the primorial covering optimum", "prize": "$1000 (for the asymptotic estimate; unclaimable by finite computation)", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "next A048670 term requires the upper-bound side: certified nonexistence over all residue choices - pruned-exhaustion receipt, not a witness", "frontier": "A048670 exact to a(64) (Bozek 2021, Google-Cloud pruned exhaustion); next term = pruned-exhaustion certificate claim", "verifier": "Lower-bound side is a POLY-TIME exact verifier: given a claimed run [u,u+V−1] and the residues, check each of the V integers is divisible by some prime ≤x — O(V·π(x)). Cheap. BUT the exact value / OEIS-table entry requires the UPPER-bound side: certify NO run of length V+1 exists over all ∏_{p≤x}p residue choices — an exhaustive/pruned nonexistence search whose cost is the whole difficulty (no compact DRAT-style certificate is standard here; correctness rests on the specialized search's completeness).", "finding": "", "erdos_url": "https://www.erdosproblems.com/687", "oeis": [ "A048670" ], "arxiv": [] }, { "id": 703, "title": "Forbidden-intersection family T(n,r) (Erdős #703)", "prize": "$250 (already awarded/solved — problem marked PROVED)", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "extremal witness has size ~2^(n-1) - not representable at any n where the answer is unknown", "frontier": "", "verifier": "Given a candidate family F, check |A∩B|≠r for all pairs: O(|F|²·n), poly in witness size. Fatal caveat: the extremal witness has size ~2^(n-1), exponential in n, so a write-down-able-and-checkable witness exists only for n≲20-22.", "finding": "", "erdos_url": "https://www.erdosproblems.com/703", "oeis": [ "A390645" ], "arxiv": [] }, { "id": 707, "title": "Sidon set extends to a perfect difference set (Erdős #707)", "prize": "$1000 (offered in Er80) — ALREADY CLAIMED; problem is DISPROVED and Lean-verified. No prize available.", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "headline DISPROVED, Lean-verified (Alexeev-Mixon, PNAS 2025); nothing movable remains", "frontier": "", "verifier": "Per fixed prime power q: check B's (q+1)q ordered differences hit each nonzero residue mod q²+q+1 exactly once (O(q²)), and check {0,1,3,11} lies in the translation×multiplier orbit of B (O(q²)). Cheap and exact — but only decides ONE modulus; the actual question ranges over infinitely many q with no upper bound.", "finding": "", "erdos_url": "https://www.erdosproblems.com/707", "oeis": [], "arxiv": [ "2510.19804", "2604.25214" ] }, { "id": 708, "title": "g(n) divisibility over windows of consecutive integers", "prize": "$100 (Erdős), lowest tier", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "no tracked record, and the upper-bound direction quantifies over unbounded sets with no finite verifier", "frontier": "", "verifier": "For a candidate lower-bound witness (A, I): exact check that no B ⊆ I of size ≤ k has ∏A | ∏B — exhaustive over 2^m subsets (m = max(A)) with trivial integer-divisibility tests. Cheap for small m, exponential as m grows. NOTE: the UPPER bound g(n)≤k has NO finite verifier — it quantifies over the infinite family of n-subsets A with unbounded max(A), with no proven a priori cap.", "finding": "", "erdos_url": "https://www.erdosproblems.com/708", "oeis": [], "arxiv": [] }, { "id": 712, "title": "Hypergraph Turán density of K_k^r (tetrahedron 5/9)", "prize": "$500", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "movable claim = an improved flag-algebra SDP dual certificate - exact-rational certificate check, heavy and specialist (LP/SDP lane)", "frontier": "5/9 <= pi(K_4^3) <= 0.561666 (Razborov flag-algebra SDP 2010); movable claim = a better SDP dual certificate", "verifier": "Poly-time exact: given an r-graph, test all C(n,k) k-subsets are non-complete → O(n^k) (O(n^4) for the tetrahedron). For an exact-value optimality claim, a SAT decision \"∃ K_k^r-free r-graph with ≥m edges\" yields a DRAT nonexistence certificate at m+1. Cheap either way.", "finding": "", "erdos_url": "https://www.erdosproblems.com/712", "oeis": [], "arxiv": [] }, { "id": 720, "title": "Size Ramsey numbers of paths & cycles", "prize": "$100 (already awarded — problem solved by Beck)", "beatable": "WALL", "board_class": "HEAVY", "board_class_reason": "each exact-value move = host graph + DRAT UNSAT certificate (upper) + enumeration receipt (lower) - certificate-tier adjudication", "frontier": "size-Ramsey exact small values (r-hat(P5,P5)=11, ...); each upper-bound move = one host + one DRAT UNSAT certificate", "verifier": "Arrowing check H→G: SAT-encode \"∃ 2-colouring of E(H) with no monochromatic G\", solver returns UNSAT (H arrows) or a witness colouring (H fails). coNP-complete (Burr). UPPER bound = one host H + a single DRAT UNSAT cert: cheap, farm-friendly, our exact pattern. EXACT value additionally needs the LOWER bound: nauty-enumerate every host with ≤m−1 edges and exhibit an avoiding colouring for each — Σ₂-style, super-exponential host count, the binding wall.", "finding": "", "erdos_url": "https://www.erdosproblems.com/720", "oeis": [], "arxiv": [] }, { "id": 1029, "title": "R(k)/(k·2^{k/2}) → ∞ (diagonal Ramsey growth)", "prize": "$100 (attached to the asymptotic theorem, which is unreachable by finite computation)", "beatable": "WALL", "board_class": "READY", "board_class_reason": "witness = 2-coloring of K43; mono-K5 count, exact, cheap (our R(5,5) verifier); a 0-defect coloring proves R(5,5) >= 44 - EinsteinArena-style market board", "frontier": "R(5,5) in [43,46]; best public K43 colorings have exactly 2 monochromatic K5s (a 0-defect witness proves R(5,5) >= 44)", "verifier": "Given an n-vertex graph G, check G contains no K_k and its complement contains no K_k. Poly-time for fixed k (clique enumeration; trivial in practice for k=5,6,7 up to n~150). Cheap, exact — same verifier class as our R(5,5) work.", "finding": "", "erdos_url": "https://www.erdosproblems.com/1029", "oeis": [], "arxiv": [] }, { "id": 1135, "title": "Collatz (3n+1) convergence", "prize": "$500", "beatable": "WALL", "board_class": "NONE", "board_class_reason": "exhaustion claim (2^71 sieve) has no certificate - adjudication = full recompute at GPU-supercomputer scale; cycle-witness side is an existence bounty with no frontier", "frontier": "", "verifier": "Trivial and exact: iterate f from a seed until it hits 1 (converged) or exceeds the tested bound; a cycle witness is verified by following the finite orbit back to its start (O(cycle length) exact integer arithmetic). Verification cost is O(1) memory per seed but the *search* is O(N) seeds — the checker is cheap, the exhaustion is the cost.", "finding": "", "erdos_url": "https://www.erdosproblems.com/1135", "oeis": [], "arxiv": [] } ], "gap_ledger": [ { "problem": 1, "oeis": "A276661", "quantity": "A276661(11) — least possible largest element of an 11-element set of positive integers whose 2^11 subset sums are pairwise distinct", "kind": "next_cell", "lower": { "value": "462", "year": 2021 }, "upper": { "value": "594", "year": 1969 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Exact terms a(0..10) = 0,1,2,4,7,13,24,44,84,161,309; a(10)=309 proven by P. W. Dyson (2025, exhaustive search, code on GitHub pwdyson/erdos_1). Conway–Guy matches a(n) exactly for all n<=10 and is conjectured optimal at n=11, which is why the upper-side witness is graded hard: recent OEIS activity (Popov Nov 2025) improved a(12)<=1157 and a(13)<=2249 but left 594 untouched. The live bracket is 462 <= a(11) <= 594. Exact closure of a(11) is a Dyson-style bounded backtracking campaign (no compact DRAT known; lower-bound claims need reproducible search transcripts). Headline $500 conjecture (N >> 2^n) is asymptotic and untouched by this cell. Jan 2026 arXiv:2601.07068 (greedy dissociated sets) checked: does not change these bounds." }, { "problem": 3, "oeis": "A003002", "quantity": "A003002(212) = r_3(212) — maximum size of a subset of {1,...,212} containing no 3-term arithmetic progression", "kind": "next_cell", "lower": { "value": "43", "year": 2026 }, "upper": { "value": "44", "year": null }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "The next-cell question is exactly whether A065825(44) <= 212 (A065825 has 43 terms, a(43)=209). Actively attacked: arXiv:2606.04016 (Jun 2026) runs a CP-SAT architecture on precisely this cell and reports a 'structural hard pocket' — integrator should check whether that paper is affiliated with our own farm before claiming the lane. Nonexistence side is a natural CNF; UNSAT+DRAT closes the cell. Sibling next cells for the same Erdős #3 family: A003003 (no 4-AP) known n<=142, A003004 (no 5-AP) n<=115, A003005 (no 6-AP) n<=147, all Cariboni b-files with extremal-set a-files — each first uncomputed cell is an analogous SAT/ILP target. The $5000 headline (divergent harmonic sum forces arbitrarily long APs) is asymptotic and untouched by any finite cell." }, { "problem": 5, "oeis": "A001223", "quantity": "S = set of limit points of (p_{n+1}-p_n)/log n — conjecture that S = [0,infinity]", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known partial results (paraphrased from erdosproblems.com/5 remarks, 2026-07-18): infinity is in S (Westzynthius 1931), 0 is in S (Goldston–Pintz–Yildirim 2009), S has positive Lebesgue measure (Erdős 1955; Ricci 1956), S contains arbitrarily large finite numbers (Hildebrand–Maier 1988), [0,c] subset of S for some c>0 (Pintz 2016), at least 1/3 of [0,infinity) lies in S and S has bounded gaps (Merikoski 2020), at least 12.5% (Banks–Freiberg–Maynard 2016). A001223 is the raw prime-gap sequence — fully computable, no uncomputed cell, no record quantity. Theorem-shaped frontier only; nothing for the records scout." }, { "problem": 9, "oeis": "A006286", "quantity": "Upper density of the set of odd numbers not representable as p + 2^a + 2^b (p prime) — is it positive?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "State of knowledge (paraphrased from erdosproblems.com/9, edited Apr 2026): Crocker (Pacific J. Math 36, 1971) proved infinitely many such odd integers, with >> log log N of them below N; Pan (arXiv:0905.3809) improved the count to >>_eps N^(1-eps) for every eps>0. The remaining gap — from exponent 1-eps to positive density — is theorem-shaped, not witness-shaped; Erdős believed covering-system arguments cannot settle it (representable numbers exist in every infinite AP). A006286 (terms 1,2,3,128,150,252,...; b-file to 10000 terms by Chai Wah Wu) is routinely extendable but extension is not a record. No bounded finite quantity with a gap." }, { "problem": 10, "oeis": "A387053", "quantity": "K* = least k (if it exists) such that every integer n >= 2 is the sum of a prime and at most k powers of 2; equivalently sup_n A387053(n)", "kind": "bounded_below_only", "lower": { "value": "4", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Existence of any finite K* is the open problem: Erdős called it 'probably unattackable'; Erdős–Graham (1980) lean toward no such k existing, so K* may be infinite (hence exact_feasibility=wall — only the lower bound is movable). Granville–Soundararajan (1998) conjecture 3 powers suffice for all odd n (hence 4 for even); Grechuk's even witness plus the abundance of A006286 suggests infinitely many even n need >= 4. Gallagher (1975): for any eps>0 some k(eps) covers a set of lower density >= 1-eps. A387053 (a(n) = least k for each n, offset 2; b-file to n=10001 by Corneth) gives the per-n table; recent related literature: Johnston–Trudgian, arXiv:2605.17825 (2026), on Linnik–Goldbach/Romanov constants. A k=4 lower-bound witness likely needs covering-congruence sieving over structured candidate families rather than brute scan — graded hard, not plausible." }, { "problem": 11, "oeis": "A377587", "quantity": "A377587(13) — smallest odd m such that m - 2^k is non-squarefree for every k = 1..13", "kind": "next_cell", "lower": { "value": "> 1125899906842624 (= 2^50)", "year": 2024 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdős #11 asks if every odd n > 1 is squarefree + a power of 2; verified to 2^50 ~ 1.13e15 (Hercher 2024; earlier Odlyzko to 1e7). A377587 is the natural record ladder: a(1..12) = 11, 29, 533, 849, 434977, ..., 60690478781437; a(13) is the first uncomputed cell. Witnesses are found by covering systems on exponents (p^2 | m - 2^k propagates along k in arithmetic progressions of the order of 2 mod p^2), the same machinery that produced a(12) — hence plausible, not easy. Exact value = bounded upward sweep continuing Hercher's method (compute-heavy but cell-shaped). Deep connection: Granville–Soundararajan (1998) tie a negative answer to Wieferich primes (A001220: only 1093 and 3511 known; OEIS records no others below ~5e17 per PrimeGrid, search later suspended). A counterexample to the conjecture itself (odd m failing for ALL k with 2^k < m) would close the problem negatively and is fully machine-verifiable — but none is expected." }, { "problem": 13, "oeis": null, "quantity": "Erdős–Sárközy f(N) = max |A|, A ⊆ {1,…,N}, with no a,b,c ∈ A (b ≠ c) such that a | (b+c) and a < min(b,c) — certified exact table for N ≤ 45, the finite side Bedert's asymptotic theorem does not give; empirical lower fence for its ineffective 'sufficiently large' threshold", "kind": "verified_range", "lower": { "value": "45", "year": 2026 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C2", "note": "Effectivization record (charter §3/WS4) — a LOWER FENCE, not a threshold location: in the computed range N ≤ 45 the last exception to f(N)=⌊N/3⌋+1 is N=17. Because Bedert's theorem (arXiv:2301.07065) proves the bound only for sufficiently large N with an INEFFECTIVE threshold, sporadic exceptions at larger N are NOT excluded; raising the fence = extending the certified table past N=45. The headline conjecture is a proven theorem (Bedert 2023), and certifying the TRUE threshold is a wall (needs an effective proof, no compact certificate) — the untabulated finite table is the content here. oeis = null: f is not in OEIS as of 2026-07-17 (checked at certification time; candidate new sequence, deliberately not submitted pending a full literature/definition cross-check — external submissions are human-sent). The stub's A002264 (⌊n/3⌋) is the comparison bound, not f." }, { "problem": 14, "oeis": "A143824", "quantity": "Size of {1..N} minus B, where B = integers representable in exactly one way as a sum of two elements of an infinite set A — conjectured >>_eps N^(1/2-eps) for every A", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Context (paraphrased from erdosproblems.com/14, edited Sep 2025): attributed to Erdős–Nathanson / Erdős–Sárközy–Szemerédi; Erdős claimed a construction with complement <<_eps N^(1/2+eps) yet with infinitely many N where the complement is >>_eps N^(1/3-eps). The only finite handle is the Erdős–Freud (1991) finite analogue: some A in {1..N} leaves fewer than 2^(3/2)*sqrt(N) integers not uniquely representable, with 2^(3/2) suggested optimal — an untabulated constant-optimization that could seed a future OEIS-style table, but no maintained table exists today. The stub's linked A143824 (max Sidon subset of {1..n}; b-file n=0..500 by Sloane; ILP formulation by Pratt in comments) is tagged only 'possible' relevance on the problem page; its next cell a(501) is a computable ILP target but belongs to the Erdős #30 Sidon lane, not #14." }, { "problem": 17, "oeis": "A038133", "quantity": "Are there infinitely many cluster primes (odd p such that every even number <= p-3 is a difference of two primes both <= p)?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Blecksmith–Erdős–Selfridge (Amer. Math. Monthly 106 (1999) 43–48) proved cluster primes are sparse: pi_C(x) <<_A x/(log x)^A for every A, and the sum of reciprocals of cluster primes converges (direction confirmed via MathWorld ClusterPrime). Elsholtz (2003) improved to << x*exp(-c(log log x)^2) for every c < 1/8 (per erdosproblems.com/17). First non-cluster prime is 97; A038133 lists non-cluster primes (b-file 10000 terms, T. D. Noe). Tao has flagged the problem as difficult on the erdosproblems page. Pure existence question — no bounded quantity with a gap; do not board." }, { "problem": 18, "oeis": "A005153", "quantity": "h(m) for practical m (fewest distinct divisors of m that always suffice to represent every 1 <= n < m) — are there infinitely many practical m with h(m) < (log log m)^O(1)? Also h(n!) < n^o(1)?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known (paraphrased from erdosproblems.com/18, edited Apr 2026): Erdős showed h(n!) < n; Vose (1985) proved infinitely many practical m have h(m) << (log m)^(1/2); the $250 reward (Erdős 1981) is for h(m) < (log log m)^O(1) infinitely often; strengthenings h(n!) < n^o(1) or (log n)^O(1) also open. A005153 (practical numbers; b-file 10000 terms, Eldar/Noe) is definitionally computable with no uncomputed cell — the sequence itself carries no record. A tabulation of h(n!) for small n would be new but is uncharted (no defended record), heavy per term, and moves no stated bound. Related problems 304, 825 on the same page." }, { "problem": 20, "oeis": "A332077", "quantity": "A332077 cell Sun(3,4) — minimal number of distinct sets of cardinality <= 3 forcing a 4-sunflower (first uncomputed cell of the sunflower-number table, in OEIS antidiagonal reading order)", "kind": "next_cell", "lower": { "value": "39", "year": 1992 }, "upper": { "value": "163", "year": 1960 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Table state (OEIS example section): row m=1 and columns n<=2 trivial; row m=2 closed form (Abbott–Hanson–Sauer); Sun(3,3)=21 proven (Erdős–Rado; 20-set extremal family in the OEIS a-file); data stops exactly at Sun(3,4). Sibling open cells with 30–50-year-old construction records: Sun(4,3) >= 55 (Abbott–Hanson–Sauer 1972, 54-set family) and Sun(3,5) >= 89 (Abbott–Hanson 1974, 88 triples), both per hub deep audit atlas/problems.json#20. Witness search is SA-seeding + ILP/MaxSAT over a swept ground set (deep atlas: hours to seed, days–weeks for fixed-g optima); records are sticky structured Exoo-class constructions — plausible, not easy. Exact value graded wall: the ground set is unbounded, so certifying the upper side needs a ground-set-size theorem, not a fixed-vertex DRAT closure. The $1000 headline (f(n,k) < c_k^n) is asymptotic and untouched; recent progress (Rao; Bell–Chueluecha–Warnke (Cs log k)^k) is all on the asymptotic upper side. Keyword 'hard,more' on the OEIS entry confirms open table status." }, { "problem": 23, "oeis": "A389646", "quantity": "A389646(24) — maximum over triangle-free graphs on 24 vertices of the minimum number of edge deletions needed to make the graph bipartite (first uncomputed term; a(23)=20 is the last known value)", "kind": "next_cell", "lower": { "value": "20", "year": 2025 }, "upper": { "value": "24", "year": 2025 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Headline: do n^2/25 deletions always suffice on 5n vertices (sharp via balanced C5 blow-up)? Best general upper: a(n) <= n^2/23.5 (Balogh-Clemen-Lidicky, arXiv:2103.14179). Ferudun (arXiv:2606.28041, Jun 2026) proved a(5k)=k^2 for all k<=40, i.e. the conjectured value holds at n=25,30,...,200 — so the live machine lane is non-multiples of 5, starting at a(24) in [20,24]. Caution: the (5,5,5,5,4) blow-up heuristic gives only 20, so a witness >20 may not exist if a(24)=20; the cell itself looks reachable by McKay-style enumeration (a(23) fell Oct 2025)." }, { "problem": 30, "oeis": "A003022", "quantity": "A003022(29) — length of the optimal (shortest) 29-mark Golomb ruler; first unproven cell after OGR-28 (a(28)=585, proved Nov 2022)", "kind": "next_cell", "lower": { "value": "586", "year": 2022 }, "upper": { "value": "623", "year": 1984 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The $1000 headline (h(N) = sqrt(N) + O_eps(N^eps) for Sidon sets) is asymptotic and not finitely computable. Every proven-optimal ruler from 10 to 28 marks matched a Singer/Bose/affine-projective construction; Rokicki-Dogon offer $250 per beaten construction length (36+ marks) and report no beats — so beating 623 is construction-defeating territory. The nonexistence side (proving 623 or anything optimal) is a distributed.net-scale exhaustion with no compact certificate (OGR-28: 8.5 years on a global grid). Alternative cheap-witness lane from atlas deep record (atlas/problems.json#30): the rational-certificate constant b in h(N) <= sqrt(N) + b*N^{1/4} + O(1), currently 0.9435 (arXiv 2607.01169, Jul 2026; prior 0.97633 AlphaEvolve-assisted 2025, 0.98183 Carter-Hunter-O'Bryant 2025) — a finite convex-QP dual certificate checkable in exact arithmetic in seconds. Companion sequences A143824 (max Sidon subset of {1..n}, b-file to 500) and A227590." }, { "problem": 36, "oeis": "A393584", "quantity": "Erdős minimum-overlap constant mu = lim M(n)/n, where M(n) = A393584(n) is the minimax, over equal partitions {X,Y} of {1..2n}, of the maximum number of pairs (x,y) with x-y equal to a fixed difference", "kind": "value_gap", "lower": { "value": "0.379005", "year": 2022 }, "upper": { "value": "0.3808594223653146192081122", "year": 2026 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "record_in_progress", "confidence": "C3", "note": "OEIS terms known to n=33 (a(33)=13; a(n)=floor(5n/13) for small n) — next uncomputed cell a(34). Bound history: trivial 1/4; Scherk 1-1/sqrt(2); Haugland 2016 upper 0.3809268534330870; AlphaEvolve [GGTW25] then TTT-Discover [YKLBMWKCZGS26] to 0.380876; project v1.2 exact-rational record 0.38085942 (tightest repaired variant 0.38085906 needs a sub-ULP admissibility repair, not admissible outright). The lower side (raising 0.379005) needs a proof, not a witness — asymmetric gap. mu itself is an analytic limit, flagged 'cannot be resolved with a finite computation' upstream. External submission of the project record is pending human review; upstream page not yet reflecting it." }, { "problem": 51, "oeis": "A002202", "quantity": "Existence of an infinite set A of totient values such that n_a/a -> infinity, where n_a is the least n with phi(n) = a (no single bounded numeric quantity attached)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Pure existence question; erdosproblems.com/51 explicitly marks it not resolvable by finite computation. Adjacent famous open: Carmichael's conjecture (A014197 contains no 1; Erdős proved one such t would force infinitely many). Both linked OEIS sequences are fully computed in their published ranges (A002202 b-file to 10000, A014197 b-file to 10000) — no uncomputed cell exists to attack." }, { "problem": 52, "oeis": "A263996", "quantity": "Integer sum-product exponent delta* = sup{delta : max(|A+A|,|AA|) >> |A|^{delta - o(1)} for all finite A subset of Z} (Erdős–Szemerédi conjecture: delta* = 2)", "kind": "value_gap", "lower": { "value": "1962/1469 (~1.33560)", "year": 2025 }, "upper": { "value": "2", "year": 1983 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "MAJOR 2026 development caught this pass: Bloom–Sawin–Schildkraut–Zhelezov [BSSZ26] (arXiv:2605.28781) DISPROVED the conjecture for A subset of R — arbitrarily large real sets with max(|A+A|,|AA|) <= |A|^{2-c} — and for small subsets of F_p and the higher-fold real variant. The integer case (this problem) remains open, and the best real/integer lower bound is Cushman's 1962/1469. Atlas deep record atlas/problems.json#52 is STALE on both counts (still cites Bloom 1270/951 and predates the refutation) — recommend an update task. A263996 exact/best-known terms with b-file to n=205." }, { "problem": 60, "oeis": "A006855", "quantity": "A006855(41) = ex(41; C4) — maximum edges in a C4-free graph on 41 vertices (first uncomputed term; exact values end at a(40)=127)", "kind": "next_cell", "lower": { "value": "132", "year": 2022 }, "upper": { "value": "133", "year": 2023 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Headline #60 is a supersaturation conjecture (any n-vertex graph with > ex(n;C4) edges contains >> sqrt(n) copies of C4; He–Ma–Yang proved it for n = q^2+q+1 with q even; upstream flags it not finitely computable) — the linked A006855 table is the machine lane, and ex(n;C4) proper is problem #765's subject. McKay marks his 41-45 lower bounds as 'good chance of being exact', so the 133-edge witness at n=41 may not exist; the wider 46-49 gaps are the honest hunting ground. Cell graded 'unknown' rather than drat-candidate: enumeration reached n=40, but the 1-edge gap at n=41 has stood 3+ years against McKay and Alekseyev, and a SAT nonexistence cert (820 edge vars, ~3*10^5 C4 clauses + cardinality 133, heavy symmetry) is unproven at this scale." }, { "problem": 68, "oeis": "A331373", "quantity": "Irrationality status of E = Sum_{n>=2} 1/(n!-1) = 1.25349875569995347... (decimal expansion = A331373)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Confirmed still open as of 2026. Active adjacent literature makes this worth a literature-watch flag rather than a compute lane: Tao's Ahmes-series irrationality results (arXiv:2406.17593, 2024) and the 2026 resolution of an Erdős–Graham rapidly-converging-series irrationality question (arXiv:2601.21442) — neither covers Sum 1/(n!-1). The linked OEIS entry is a decimal expansion, so there is no uncomputed cell in the gap-map sense." }, { "problem": 77, "oeis": "A059442", "quantity": "C = lim_{k->infinity} R(k)^{1/k}, the diagonal Ramsey growth base (the limit's existence is itself open; bounds below are for liminf/limsup)", "kind": "value_gap", "lower": { "value": "sqrt(2) (~1.41421)", "year": 1947 }, "upper": { "value": "3.7992", "year": 2024 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "$250 for the value; the same page records Erdős's separate $100 offer for mere existence of the limit (and joke prizes for nonexistence). Erdős speculated the value might be 2 with 'no real evidence'. Simpler/multicolor 4-c proof: Balister et al. [BBCGHMST24]. Upstream explicitly: not resolvable by finite computation. Deep record atlas/problems.json#77 concurs (wall; our SAT+DRAT/MIS tooling mechanism-matches only the analytically disconnected feeder cells). Its '3.8' upper is consistent with the page's 3.7992." }, { "problem": 78, "oeis": "A059442", "quantity": "Constructive exponential Ramsey lower bound: an explicit family certifying R(k) > C^k for some C > 1 — equivalently explicit n-vertex graphs with clique and independence number O(log n)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "$100 prize. Erdős's probabilistic R(k) >> k 2^{k/2} is nonconstructive; the ask is an explicit C^k family. Distinct from #77 (which asks the VALUE of lim R(k)^{1/k}) despite the shared A059442 link — verified both pages directly to disambiguate. Page last edited 2026-01-23; upstream flags it not resolvable by finite computation. No OEIS cell to attack (A059442's own frontier cells like R(5,5) belong to #77's feeder discussion and to the small-Ramsey lane already covered by project campaigns)." }, { "problem": 82, "oeis": "A394563", "quantity": "A394563(6) — least N such that every graph on N vertices has an induced regular subgraph of order ≥ 6", "kind": "bounded_below_only", "lower": { "value": "21", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Witness verification is trivial; beating the bound means out-searching McKay's 2026 computation. Exact value needs a universal upper-bound proof — wall." }, { "problem": 85, "oeis": "A006672", "quantity": "A006672(39) = R(C4, K_{1,39}) — first literature-open cell of the C4-vs-star Ramsey table (the finite shadow of #85's min-degree function f(n); R(C4,K_{1,n}) = min{m : f(m) <= m-n})", "kind": "next_cell", "lower": { "value": "46", "year": 2015 }, "upper": { "value": "47", "year": 1975 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Headline #85 (is f(n+1) >= f(n) for all large n?) is explicitly not finitely resolvable (erdosproblems.com/85); known f(n) = (1+o(1))sqrt(n). The gap-shaped surface is the shared A006672 table: OEIS is stale (11 terms) but literature is complete through n=38 (Boza 2024 arXiv:2409.12770 closed the last 8 values <= 38), so n=39 is the first open cell. CAUTION: this exact cell caused this project's 07-16 retraction; the reconciled state is OPEN {46,47}. Project SAT deciders on the (46, min-deg-7) instance were killed after 5.5h with no result (2026-07-17) — graded beyond current tools. Table also cross-linked to hub #552/#242 lane." }, { "problem": 86, "oeis": "A245762", "quantity": "A245762(7) = ex(Q_7, C4) — maximum edges in a C4-free subgraph of the 7-dimensional hypercube (first uncomputed term; exact known through ex(Q_6,C4)=132)", "kind": "next_cell", "lower": { "value": "304", "year": 2026 }, "upper": { "value": "328", "year": 1994 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "$100 prize attaches only to the asymptotic Turan-density conjecture (density -> 1/2?), pinned in [1/2, 0.60318] (lower Brass-Harborth-Nienborg 1995; upper Baber flag algebras, per erdosproblems.com/86) — unclaimable by finite computation. Finite frontier: lower record 304 was just set (May 2026) by heavy two-phase simulated annealing with Aut(Q7) diversification, so beating it with the same tools has no edge (hub deep record grades wall-ish). The unattempted novel target is UNSAT+DRAT at 305 edges (448 edge vars, 672 four-cycle clauses, cardinality >= 305, lex-leader symmetry breaking over |Aut(Q7)|=645120) => first proven term past n=6; instance is small but hardness genuinely unknown (paper's ILP failed on Q7). Q8: lower 680 (same paper), computational evidence for equality, almost certainly a wall." }, { "problem": 87, "oeis": "A059442", "quantity": "inf{R(G) : chi(G)=k} relative to the diagonal Ramsey number R(k), as k -> infinity — whether R(G) > (1-eps)^k R(k) (or > c*R(k)) for every G of chromatic number k", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "erdosproblems.com/87 marks this not resolvable by finite computation. Known: original conjecture R(G) >= R(k) fails at k=4 (Faudree-McKay 1993, pentagonal wheel R=17 vs R(4)=18); trivial for eps >= 3/4 since R(k) <= 4^k; Wigderson observed R(G) >> 2^{k/2} for chi(G)=k, matching best-known diagonal lower bounds — so progress is tied to the (asymptotic) diagonal Ramsey problem itself (cf. Campos-Griffiths-Morris-Sahasrabudhe 2023 exponential improvement). OEIS link A059442 (the R(n,k) table, flagged 'possible' by the hub) is the wrong record surface for THIS problem: its famous open cells (e.g. R(5,5) in [43,46], Angeltveit-McKay) belong to hub #77/#78/#545/#986, and this project's R(5,5) witness hunt is already CLOSED-published." }, { "problem": 89, "oeis": "A131628", "quantity": "A131628(7) = g(7) — maximum number of points in a planar 7-distance set (first open cell; controls the first genuinely open cell A186704(17) of the min-distinct-distances table)", "kind": "next_cell", "lower": { "value": "16", "year": 2013 }, "upper": { "value": "36", "year": 1984 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Headline #89 ($500) is the asymptotic distinct-distances conjecture (>> n/sqrt(log n)); Guth-Katz 2015 proved >> n/log n — not finitely resolvable. Finite tables: g(k) known only for k<=6: 3,5,7,9,12,13 (Erdos-Fishburn 1996; g(6)=13 Wei 2012). Librarian note: A186704 (OEIS stops at a(13)=6) extends by pure literature synthesis — a(14)=a(15)=a(16)=7 (>= 7 since g(k)<=13 for k<=6; <= 7 via 14/15/16-point subsets of the 16-point 7-distance set); the first genuinely open cell is A186704(17), which equals 7 iff g(7) >= 17. Erdos-Fishburn conjecture: for k>=7 all maximum k-distance sets are lattice-similar; Ahmed-Snevily present data suggesting it may be false. Upper 36 is the generic s-distance bound; no sharper published planar bound for k=7 verified." }, { "problem": 91, "oeis": "A186704", "quantity": "Number of pairwise non-similar n-point configurations achieving the minimum number of distinct distances (question: >= 2, and probably many, for all large n)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "erdosproblems.com/91 marks this not resolvable by finite computation. Known small-n structure (paraphrased): n=3 equilateral triangle unique; n=4 has two non-similar minimizers (square; two glued equilateral triangles); n=5 regular pentagon unique — published proof by Kovacs 2024 (settling Erdos's 'colleague from Zagreb' anecdote); Erdos (1987) claimed >= 2 non-similar examples for 6 <= n <= 9. The page notes unincorporated comment claims of partial results/path to solution — worth re-checking on revisit. Shares the A186704 table with #89; see that entry for the live finite cell." }, { "problem": 101, "oeis": "A006065", "quantity": "A006065(21) — maximum number of lines containing exactly 4 points among 21 points in the plane (orchard problem, 4-per-row; first uncomputed cell, exact values known through a(20)=23)", "kind": "next_cell", "lower": { "value": "23", "year": 2010 }, "upper": { "value": "35", "year": null }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The $100 headline (number of 4-point lines is o(n^2) given no 5 collinear) is asymptotic and cannot be settled by one configuration. Finite surface: exact optima stuck at n=20 since ~2010; a(22) >= 28 (Du, OEIS illustration; also Pegg 2020 construction) but no verified a(21) lower bound better than monotone 23 was found. Hub deep record grades the exact direction WALL: order-type enumeration plus existsR-realizability (stretchability) — not SAT/DRAT-shaped; and lower-bound nudges must out-produce algebraic cubic-curve constructions. Integer-coordinate variant A172992 (known to n=27, Kamenetsky 2012) is provably below the real optimum for several n." }, { "problem": 104, "oeis": "A003829", "quantity": "A003829(9) — maximum number of distinct unit circles each passing through >= 3 of 9 points in the plane (first uncomputed cell; exact known through a(8)=16, Harborth-Mengersen 1986)", "kind": "next_cell", "lower": { "value": "16", "year": 1986 }, "upper": { "value": "24", "year": 1986 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "$100 headline is asymptotic: is the number of rich unit circles o(n^2)? Known (per erdosproblems.com/104, paraphrased): >> n trivially (Erdos 1981), >> n^{3/2} construction (Elekes 1984, possibly the right order); upper O(n^2), refined to n(n-1)/3 (Harborth-Mengersen 1986); Erdos offered 100 GBP for deciding O(n^{3/2}) in general position. The finite cell a(9) appears untouched since 1986 (OEIS 'more', no post-1986 refs found) — an 18-dimensional continuous search plus exact certification is a genuine, low-competition scout target; exact determination of a(9) would need real-algebraic case analysis (graded unknown)." }, { "problem": 107, "oeis": "A000051", "quantity": "ES(7) = f(7) — least N such that every N points in general position (no 3 collinear) contain a convex 7-gon; first unknown Erdos-Szekeres value (f(3..6) = 3, 5, 9, 17; conjecture f(n) = 2^{n-2}+1 = A000051(n-2), so f(7) = 33)", "kind": "next_cell", "lower": { "value": "33", "year": 1935 }, "upper": { "value": "127", "year": 2005 }, "witness_side": "lower", "witness_feasibility": "historic", "status": "open", "confidence": "C3", "note": "The Happy Ending problem — flagship $500 target; settling f(7) would be the first new exact value since f(6)=17 (Szekeres-Peters 2006; SAT-reproven in 8.53 CPU-sec by Heule-Scheucher 2024). The proof side f(7) <= 33 is DRAT-shaped in principle (signotope encoding + forbid convex 7-subsets) but currently intractable: Bogdan's Dec-2025 16.67M-clause attack closes only anchored subfamilies with heavy-tailed runtimes to 2.28M CPU-sec per configuration, and the encoding is owned by the world's leading SAT-geometry groups (Heule, Scheucher, Maric) — no edge for a 100-core farm (hub deep record verdict WALL). Falsification search targets an object believed nonexistent." }, { "problem": 131, "oeis": "A068063", "quantity": "A068063(101) — maximum size of a nondividing subset of {1,...,101} (no element divides the sum of any nonempty subset of the others); first cell beyond the b-file, which ends at a(100)=8", "kind": "next_cell", "lower": { "value": "8", "year": null }, "upper": { "value": "9", "year": 1999 }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Headline #131 asks the growth of F(N): Erdos's original question (F(N) > N^{1/2-o(1)}?) is answered NO — F(N) <= N^{1/4+o(1)} via the non-averaging-set link (Pham-Zakharov 2024), against F(N) >> N^{1/5} (Csaba construction, in ELRSS99) and Straus's exp((sqrt(2/log 2)+o(1)) sqrt(log N)); correct exponent open (asymptotic, not machine-movable). The finite table is the scout surface: branch-and-bound over {1..N} extends A068063 cheaply (jumps to size 8 at n=65; size-9 threshold unknown — locating min{N : F(N)=9} is a clean, ownable record). Cross-ref Erdos #13 (Erdos-Sarkozy) and #186 (non-averaging), already worked in-project." }, { "problem": 138, "oeis": "A005346", "quantity": "A005346(7) = W(2,7) — least N such that every 2-coloring of {1..N} contains a monochromatic 7-term arithmetic progression (first uncomputed van der Waerden number; W(2,3..6) = 9, 35, 178, 1132)", "kind": "bounded_below_only", "lower": { "value": "> 3703", "year": 2012 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The $500 headline (W(k)^{1/k} -> infinity) needs an asymptotic theorem (cf. Green 2021 w(3,k) lower bounds; sharpened by arXiv:2111.01099) — unreachable by witness. Finite surfaces per hub deep record: (1) exact W(2,7) via UNSAT+DRAT at N ~ 10^3-10^4 is astronomically infeasible (W(2,6)=1132 already needed clusters+FPGAs, Kouril-Paul 2008) — R(5,5)-class wall; (2) the lower-bound record is a live SLS/SAT search with a trivial verifier but is held by purpose-tuned specialist solvers (cyclic zippers, Heule-style constructions, Monroe's DDFW linear-weight-transfer); 14 years unbeaten, no verifier moat. Related off-diagonal cells (W(2,8) >= 11495, W(2,9) >= 41265, W(2,10) >= 103474) are equally specialist-held." }, { "problem": 141, "oeis": "A006560", "quantity": "A006560(7) — smallest starting prime of 7 consecutive primes in arithmetic progression (CPAP-7)", "kind": "bounded_above_only", "lower": null, "upper": { "value": "71137654873189893604531", "year": 2022 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Only one CPAP-7 known; any smaller one is a genuine record with trivial verification. The exact minimum needs an infeasible ~1e21 exhaustive sweep." }, { "problem": 142, "oeis": "A003002", "quantity": "A003002(212) — r_3(212), size of the largest subset of {1..212} with no 3-term arithmetic progression (first uncomputed cell; exact table verified to n=211)", "kind": "next_cell", "lower": { "value": "43", "year": 2018 }, "upper": { "value": "44", "year": 2018 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Run structure (a(209..211)=43,43,43; recent runs of length 3-10) suggests a(212)=43, so the likely record is the closure certificate, not a size-44 set. The named $10000 problem (asymptotic formula for r_k(N)) is a wall: best upper bounds Kelley-Meka 2023 (k=3), Green-Tao 2017 (k=4), Leng-Sah-Sawhney 2024 (k>=5), per erdosproblems.com/142 (page dated 2026-04-04). Companion next cells: A003003(143) [r_4, table to 142], A003004(116) [r_5, to 115], A003005(148) [r_6, to 147], all Cariboni 2018 with witness sets published on OEIS." }, { "problem": 145, "oeis": "A005117", "quantity": "alpha* — supremum of moment exponents alpha for which lim_{x->inf} (1/x) * sum_{s_n<=x} (s_{n+1}-s_n)^alpha over gaps of squarefree numbers is PROVEN to exist", "kind": "bounded_below_only", "lower": { "value": "3.75", "year": 2023 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Granville 1998: the full conjecture follows from the ABC conjecture. The linked A005117 is just the squarefree sequence itself — no frontier cell. Not a records-lane target; recorded for gap-map completeness. erdosproblems.com/145 page last edited 2025-10-19." }, { "problem": 148, "oeis": "A006585", "quantity": "A006585(9) — number of Egyptian-fraction solutions to 1 = 1/x_1 + … + 1/x_9", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "a(8) = 151182379 (Dethridge 2004); a(9) expected ~1e12–1e13 solutions; 22 years stuck. Recorded so the lane knows to skip count-shaped problems." }, { "problem": 155, "oeis": "A003022", "quantity": "A003022(29) — length of the optimal (shortest) 29-mark Golomb ruler; first unproven cell after distributed.net OGR-28 closed a(28)=585 in Nov 2022", "kind": "next_cell", "lower": { "value": "586", "year": 2022 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Relevance to #155 (is F(N+k) <= F(N)+1 for large N, F = A143824 max Sidon subset size): F is determined by the optimal-ruler table — F jumps exactly at A227590(n)=a(n)+1 — so pinning a(29) extends the verified range of the F(N+k)<=F(N)+1 conjecture beyond N=586. Bloom remark: conjecture may even hold with k ~ eps*N^(1/2). No ruler with >16 marks has ever beaten the finite-field constructions (Rokicki-Dogon hypothesis A, with a $250 bounty for orders 36..40000); OGR-28 consumed ~8 years of distributed volunteer CPU and OGR-29 was never launched — exact closure is a wall for one-box compute." }, { "problem": 156, "oeis": "A382397", "quantity": "A382397(66) — minimum size of a maximal Sidon subset of {1..66} (first uncomputed cell; OEIS table computed through n=65, added 2025)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Data: a(n)=6 for all computed 43<=n<=65; monotonicity is empirically clean but unproven, so no rigorous lower/upper carried. Exhaustive closure of a(66) (and pushing to n~100) is a bounded search (C(66,6)~9e7 candidates with heavy Sidon pruning) — hours on one box; OEIS comment explicitly notes there are too few terms to pin the growth pattern, so new cells have real value. Asymptotic open content (Erdős-Sárközy-Sós 1994): greedy gives >> N^(1/3); Ruzsa 1998 (A small maximal Sidon set, Ramanujan J.) constructed maximal Sidon sets << (N log N)^(1/3); closing the N^(1/3) vs (N log N)^(1/3) factor is the prize. See also #340." }, { "problem": 165, "oeis": "A000791", "quantity": "R(3,10) = A000791(10) — the smallest open Ramsey number in the R(3,k) family (table exact through R(3,9)=36)", "kind": "value_gap", "lower": { "value": "40", "year": null }, "upper": { "value": "41", "year": 2025 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Freshness-checked 2026-07-18: 40 <= R(3,10) <= 41 is current; Angeltveit's bound is now peer-reviewed (EJC Nov 2025). Deciding 40 vs 41 requires ruling out all (3,10)-graphs on 40 vertices — Angeltveit-scale gluing computation, beyond one-box scope. The named $250 problem is the asymptotic: R(3,k) = c*k^2/log k with c in [1/2, 1]; lower c>=1/2 Hefty-Horn-King-Pfender 2025 (chain: Kim 1995 1/162, Bohman-Keevash & Pontiveros-Griffiths-Morris 1/4, Campos-Jenssen-Michelen-Sahasrabudhe 1/3), upper (1+o(1)) Shearer 1983; both 2025 papers conjecture c=1/2. Not machine-movable on the asymptotic side; the finite cell is the only witness surface." }, { "problem": 168, "oeis": "A386439", "quantity": "The Graham-Spencer-Witsenhausen limit: lim_n (largest weakly-triple-free subset of the first n 3-smooth numbers)/n = maximal density of {x,2x,3x}-free sets = 0.80096575500655898909... (A386439)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Remaining open content is qualitative: is the constant irrational (open per A386439 comment), plus density/upper-density analogues for infinite sets (erdosproblems.com/168, page dated 2026-03-23). The finite tables A057561/A094708/A004059 are generated by an efficient algorithm (Eberhard's SageMath script), so there is no expensive frontier cell. Not a records-lane target." }, { "problem": 169, "oeis": "A005346", "quantity": "f(3) — supremum of sum_{a in A} 1/a over all 3-AP-free sets A of positive integers (Erdős-Graham); finite because reciprocal-sum divergence forces 3-APs (Bloom-Sisask 2020)", "kind": "bounded_below_only", "lower": { "value": "3.00849", "year": 1984 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Record is 40+ years old, hence graded hard, but the attack surface just changed: Walker [Wa25] pushed f(4) >= 4.43975 via the Kempner reduction, and the (b,S) search space is machine-enumerable with cheap rigorous evaluation — a genuine hunt candidate. Context: Berlekamp 1968 f(k) >= (log2/2)k; Gerver 1977 f(k) >= (1-o(1))k*log k; trivially f(k)/log W(k) >= 1/2 and any constant > 1/2 is open — this is the link to the stub's OEIS A005346 = van der Waerden numbers W(2,n) (known through W(2,6)=1132, Kouril-Paul 2008; W(2,7) open, lower-bound colorings are a separate SAT-witness surface). No explicit numeric upper bound for f(3) is published (finiteness only), hence bounded_below_only." }, { "problem": 170, "oeis": "A046693", "quantity": "A046693(214) — minimal number of marks in a sparse ruler / restricted difference basis for {0..214} (first cell beyond Ed Pegg Jr's verified table n=0..213)", "kind": "next_cell", "lower": { "value": "22", "year": null }, "upper": { "value": "26", "year": null }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The named problem (Erdős-Gál 1948, answering Rédei) is the limit constant lim a(n)/sqrt(n), proven to exist and known to lie in [1.56, sqrt(3)=1.73205...]: lower Leech 1956, upper Wichmann 1963; Pegg's computations [Pe20] give evidence the Wichmann upper bound is the truth. Finite witnesses cannot move the limit's bounds — but each new exactly-closed cell extends the excess-conjecture evidence (cf. A308766 divergence list, A326499 best-known solutions). Beware non-monotonicity: a(137) < a(136) and a(212)=25 < a(211)=26 (b-file), so only the +1-step and counting bounds are rigorous. Exhaustive closure of one cell at n=214 (choosing ~25 of 215 positions) is within reach of the same desktop-class search that produced the 0..213 table." }, { "problem": 171, "oeis": "A156989", "quantity": "c_7 — largest line-free subset of {1,2,3}^7 (density Hales–Jewett, = A156989(7))", "kind": "value_gap", "lower": { "value": "1302", "year": 2010 }, "upper": { "value": "1386", "year": 2010 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "16-year bracket; +1 on the lower bound would be the first movement since Polymath1. Exact value needs an optimality certificate that does not exist compactly." }, { "problem": 183, "oeis": "A003323", "quantity": "R(3,3,3,3) — 4-color Ramsey number of the triangle (= A003323(4))", "kind": "value_gap", "lower": { "value": "51", "year": 1971 }, "upper": { "value": "62", "year": 2004 }, "witness_side": "lower", "witness_feasibility": "historic", "status": "open", "confidence": "C3", "note": "50-year gap; even +1 on the lower bound would be the first movement since ~1971. DS1 rev.17 still lists 51–62; Wesley arXiv:2509.03784 (2025) improves other multicolor numbers, not this one. OEIS deleted the value in Aug 2021 'since not known'." }, { "problem": 185, "oeis": "A003142", "quantity": "c'_7 — 7-dimensional Moser number (largest subset of [3]^7 with no geometric 3-in-line)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "c'_6 = 353 (Polymath 2010); published c'_7 bounds NOT yet sourced here — a miner must cite current best lower/upper before any record attempt." }, { "problem": 193, "oeis": "A231255", "quantity": "A231255(7) — least t such that every t-step North-East lattice path (steps (0,1)/(1,0)) contains 7 collinear points (Gerver–Ramsey NE variant tracked by the hub; the hub statement itself is the Z^3 S-walk question)", "kind": "next_cell", "lower": { "value": ">= 328", "year": 2025 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "a(1..6) = 0,1,4,9,29,97; Barnoff–Bright enumerated all avoiding paths for k<=6 by SAT, so the k=7 frontier is a live SAT lane (record set Nov 2025 — beating it needs comparable incremental-SAT effort, not hours). No numeric upper bound: von Brömssen (May 2026) gives a(7) <= A380991(6) but A380991 (polyomino version) is only computed to n=4. Exact a(7) = UNSAT of avoidance at the threshold length — clean DRAT target in principle, scale unknown." }, { "problem": 200, "oeis": "A005115", "quantity": "A005115(27) — smallest possible last term of an arithmetic progression of 27 primes", "kind": "next_cell", "lower": { "value": "> 12783396861134173", "year": 2026 }, "upper": { "value": "696112717486210091", "year": 2019 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "FRESH state: minimal-end table now proves a(1..26) all minimal (k=23 Perrenet 2023, k=24 Petukhov Nov 2025, k=25 and k=26 Perrenet Feb 2026), so the OEIS row is fully certified and a(27) is the genuine next cell with a 54x-wide two-sided window. Finding any smaller AP27 took PrimeGrid 3+ years of distributed GPU search to find even one — grade hard. Exact a(27) is structurally a bounded exhaustive search below 6.97e17 and the Perrenet minimality pipeline just reached k=26, but expect CPU-decades not GPU-hours. Erdős #200 itself (is longest prime-AP in [1,N] of length o(log N)?) is asymptotic; PNT gives <= (1+o(1))log N; the A005115 table is the machine-record content." }, { "problem": 201, "oeis": "A003002", "quantity": "A003002(212) — maximum size of a subset of {1,...,212} containing no 3-term arithmetic progression", "kind": "next_cell", "lower": { "value": "43", "year": null }, "upper": { "value": "44", "year": null }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Width-1 cell: SAT/ILP with 212 boolean vars, one clause per 3-AP (~11k), cardinality >= 44 — either a witness (44) or an UNSAT/DRAT certificate (43). Grade plausible not open-easy: boundary cells beyond Cariboni's dedicated exhaustive search (stopped at 211) historically resist. Sibling next cells for the same hub problem: A003003(143) (4-AP-free, b-file to 142), A003004(116) (5-AP-free, to 115), A003005(148) (6-AP-free, to 147), all Cariboni. Record-side sequence A065825 known to n=43 (=209). The asymptotic content (Behrend lower, Bloom–Sisask n/(log n)^{1+c}) is not machine-closable." }, { "problem": 208, "oeis": "A076259", "quantity": "A020753(19)/A020754(19) — next record gap between consecutive squarefree numbers (current record: gap 19, from 125781000834058567 to 125781000834058586)", "kind": "next_cell", "lower": { "value": "19", "year": 2019 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Records occur out of order (first gap-12 precedes first gap-11 — McNew, OEIS A020754), so the table is 'successive increasing gaps'. Extending it needs exhaustive sieving beyond ~1.26e17 (and likely beyond the ~1e18 extent of MOST 2019) — big distributed compute, hence hard. CRT gives arbitrarily large gaps at astronomical heights, so a bare large gap without sieve coverage is NOT a record witness. Theory frontier (paraphrased from erdosproblems.com/208): Erdős 1951 lower (pi^2/6)(log s)/(loglog s) infinitely often — conjectured sharp; upper s^{1/5-c} (Pandey 2024, improving Filaseta–Trifonov 1992); ABC implies O(s^eps) (Granville 1998)." }, { "problem": 218, "oeis": "A064113", "quantity": "Densities/infinitude of consecutive-prime-gap patterns: {n : d_{n+1} >= d_n} and {n : d_{n+1} <= d_n} each conjectured density 1/2, plus infinitude of d_{n+1} = d_n (A064113)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "The run version (d_n = d_{n+1} = ... = d_{n+k} solvable for all k) is equivalent to k consecutive primes in AP — that record lane lives at hub #141 (already seeded, CPAP records). Banks (2023) gives a heuristic for the density-1/2 statement conditional on a quantitative prime-tuples conjecture (paraphrased from erdosproblems.com/218, page edited Jan 2026). Empirical density tracking is possible but bounds nothing." }, { "problem": 222, "oeis": "A256435", "quantity": "A357018(44) — next record run of consecutive integers not expressible as a sum of two squares (current record: 106 consecutive, following A297350(43) = 99951600293)", "kind": "next_cell", "lower": { "value": "106", "year": 2022 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Both records sequences carry keyword 'more' and stop near 1e11 — a segmented sieve marking a^2+b^2 (or multiplicative sieve via primes 3 mod 4) to 1e12–1e13 on one GPU box is a days-scale record hunt; the most compute-accessible lane in this chunk. Finite records do NOT move the asymptotic frontier: limsup gap/log n_k >= 0.868... (Dietmann–Elsholtz–Kalmynin–Konyagin–Maynard, IMRN 2022, improving Richards 1982's 1/4), best upper O(n_k^{1/4}) (Bambah–Chowla 1947), Erdős 1951 lower log/sqrt(loglog) i.o. — numeric facts paraphrased from erdosproblems.com/222." }, { "problem": 233, "oeis": "A074741", "quantity": "Asymptotics of A074741 (partial sums of squared prime gaps): Heath-Brown's conjecture sum_{p_n <= x} (p_{n+1}-p_n)^2 ~ 2 x log x (stub's Lean form: first N gaps ~ N (log N)^2, equivalent via p_N ~ N log N)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Heath-Brown 1982 (Acta Arith. 41) ties the conjecture to pair correlation of zeta zeros; Wolf's variant 2 x^2/pi(x) is also on the OEIS entry. Cramér-class: even RH-conditional results are far from the conjectured asymptotic. No gap-map frontier; keep as librarian/context entry only." }, { "problem": 236, "oeis": "A109925", "quantity": "A109926(21) — least n whose number of representations n = p + 2^k (p prime, 2^k < n) is exactly 21; records-of-f lane for f = A109925, the quantity Erdős conjectured to be o(log n)", "kind": "next_cell", "lower": { "value": "> 1e10", "year": 2009 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Search accelerators from OEIS comments: known terms n>4 divisible by 15 (conjectured 30 | a(n) beyond); segmented prime sieve + 2^k shift counting reaches 1e11–1e12 on one box in days. Once any n0 with 21 reps is found, exact a(21) is a bounded exhaustive count over (1e10, n0] — hence cell. FRESHNESS caveat: the 1e10 floor is 2009-era; re-check for extensions before spending compute. Companion verified-range fact for the same hub problem: A039669 = {4,7,15,21,45,75,105} (m - 2^k prime for ALL k) has no further terms below 2^120 (M. Alekseyev 2011, OEIS) and Erdős conjectured the list complete (cf. hub #1142). Theory: Erdős 1950 proved f(n) >> loglog n infinitely often (paraphrased from erdosproblems.com/236)." }, { "problem": 241, "oeis": "A387704", "quantity": "A227358(11) — shortest 11-mark B3 ruler (11 integers 0 = m1 < ... < m11 with all triple sums, repetition allowed, distinct); equivalently the n where A387704 first reaches 11", "kind": "next_cell", "lower": { "value": ">= 310", "year": 2013 }, "upper": { "value": "445", "year": 2013 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "record_in_progress", "confidence": "C3", "note": "Hub #241's own conjecture (f(N) ~ N^{1/3}, Bose/Guy C11) is asymptotic and not machine-closable. Its stub sequence A387704 (b-file 0..150, max value 8) was closed as a pure crossref: A387704(n) = max{k : A227358(k) <= n-1}, 0/151 mismatches, first reaches 9 at n = 209 (hub librarian, PR #80). The only open computational content is A227358(11), which is farm-owned under hub #41 (witness search < 445 running). Tromp 2013 estimated many CPU-years to exhaust [310,439] with his C code; 13 years without improvement — grade hard despite trivial verification. O'Bryant arXiv:2308.12406 (2023) covers thick B_h constructions." }, { "problem": 242, "oeis": "A073101", "quantity": "Least counterexample n > 2 to the Erdős–Straus conjecture 4/n = 1/x + 1/y + 1/z (conjectured nonexistent); tracked as the verified-range frontier N with the conjecture confirmed for all n <= N", "kind": "verified_range", "lower": { "value": "1e18", "year": 2025 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Falsifiable: a single counterexample n would close the problem, but conjecture is believed true, and truth cannot be established by finite computation. Partial results (per erdosproblems.com/242): Mordell 1969 reduces to 6 classes mod 840; Terzi 1971 to 198 classes mod 120120; Vaughan 1970: exceptions in [1,x] <= x*exp(-c(log x)^{2/3}); Elsholtz–Tao 2013: sum of solution counts over primes <= N is N(log N)^{2+o(1)}, f(p) <= p^{3/5+o(1)}. CAUTION: Bradford, arXiv:2602.11774 (Feb 2026, 4KB, v1) claims a complete solution — unreviewed, and erdosproblems.com still lists the problem OPEN as of its 2026-05-07 edit; do not treat as closed. A073101 counts solutions with x 1 and rational sum of reciprocals must eventually satisfy the Sylvester recurrence a_n = a_{n-1}^2 - a_{n-1} + 1", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "erdosproblems.com/243 (edited 2026-01-21) lists: Erdős–Straus 1964 [ErSt64] proved a limsup criterion any non-recurrence rational-sum sequence must satisfy; Duverney 2001 [Du01] proved the variant where sum(a_{n+1}/a_n^2 - 1) converges: rational sum iff eventually Sylvester recurrence. A000058 = Sylvester's sequence 2,3,7,43,1807,... (greedy Egyptian expansion of 1); OEIS b-file has 13 terms (doubly exponential growth)." }, { "problem": 249, "oeis": "A256936", "quantity": "No bounded quantity — is sum_{n>=1} phi(n)/2^n irrational? (constant = 1.36763080198502235... = A256936)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "No partial results listed on erdosproblems.com/249 (edited 2025-09-28); one user flags it 'difficult'. OEIS gives alternative series forms (Eldar: equals 1/2 + sum mu(k)/(2^k-1)^2, Mar 2026 comment). Source: Erdős–Graham 1980 p.61." }, { "problem": 251, "oeis": "A098990", "quantity": "No bounded quantity — is sum_{n>=1} p_n/2^n irrational, p_n the n-th prime? (constant = 3.67464396601132877... = A098990)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Closest known result (erdosproblems.com/251, edited 2025-09-28): Erdős 1958 [Er58b] proved sum p_n^k/n! is irrational for every k >= 1 — the factorial-denominator analogue; the 2^n version is open. Erdős [Er88c] further conjectured sum p_n^k/2^n irrational for all k, and irrationality of sum p_n/(g_1...g_n) whenever g_n >= 2, g_n = o(p_n) (g_n = p_n + 1 shows a growth condition is needed)." }, { "problem": 252, "oeis": "A227988", "quantity": "No machine-bounded quantity — irrationality of sum_{n>=1} sigma_k(n)/n! for each fixed k >= 1; the proof frontier stands at k = 4, first open case k = 5", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Frontier (erdosproblems.com/252, edited 2026-01-22): unconditionally irrational for 1 <= k <= 4 — k=1,2 Erdős 1952 [Er52] (cf. Erdős–Kac Monthly Problem 4518, solution Breusch 1954, per OEIS A227988 comment); k=3 independently Schlage-Puchta 2006 [ScPu06] and Friedlander–Luca–Stoiciu 2007 [FLC07]; k=4 Pratt 2022 [Pr22]. Irrational for ALL k >= 1 conditional on Schinzel's hypothesis (Schlage-Puchta) or Dickson's conjecture (FLS). Guy UPINT problem B14." }, { "problem": 271, "oeis": "A005487", "quantity": "No bounded quantity — explicit description / growth rate of the greedy 3-AP-free (Stanley) sequence A(4) = S(0,4) = A005487 and its family A(n); conjectured dichotomy: growth k^{log_2 3} vs k^2/log k", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Per erdosproblems.com/271 (edited 2026-01-20): A(1), A(3^k), A(2*3^k) have explicit base-3 characterizations (Odlyzko–Stanley 1978); Odlyzko–Stanley conjectured every Stanley sequence eventually grows like k^{log_2 3} or k^2/log k; no sequence is proven to have the second rate; Lindhurst 1990 data suggests A(4) does. Moy 2011: a_k <= (1/2+eps)k^2 eventually; explicit version a_k <= (k-1)(k+2)/2 + n (van Doorn & Sothanaphan, site comments). Fresh: Sothanaphan arXiv:2512.11983 (Dec 2025) argues irregular Stanley sequences plausibly do NOT have Theta(k^2/log k) growth — the conjectured dichotomy itself is in doubt. Not formalized in Lean." }, { "problem": 291, "oeis": "A110566", "quantity": "No bounded quantity — writing H_n = a_n/L_n with L_n = lcm(1..n): do gcd(a_n, L_n) = 1 and gcd(a_n, L_n) > 1 each occur for infinitely many n? (gcd(a_n, L_n) = A110566(n))", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Per erdosproblems.com/291 (edited 2026-01-12): the gcd>1 direction is effectively resolved — site-recorded observation (Steinerberger): if the leading base-p digit of n is p-1 then p | gcd; exact criterion: p | gcd iff p divides the numerator of H_k where k = leading digit of n base p. The gcd=1 direction is the open part; heuristic count ~ x/log x of such n <= x (Shiu preprint 2016 [Sh16]), so infinitely many expected but density zero. Wu–Yan 2022 [WuYa22]: conditional on Q-linear independence of {1/log p} (follows from Schanuel), the gcd>1 set has upper density 1. A110566 b-file to n=10000 (Heinz)." }, { "problem": 295, "oeis": "A192881", "quantity": "A192881(17) — minimum number of terms in an Egyptian fraction decomposition of 1 with least denominator 17 (finite proxy table for k(N) of #295); a(1..16) known, a(16) = 30", "kind": "next_cell", "lower": { "value": "29", "year": 2026 }, "upper": null, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Problem #295 asks whether k(N) - (e-1)N -> infinity; Erdős–Straus 1971 [ErSt71b] proved -c < k(N) - (e-1)N << N/log N (per erdosproblems.com/295, edited 2025-10-01). (e-1)*17 ~ 29.2, consistent with the harmonic lower bound 29. Table is live: Alekseyev added a(11)-a(15) with explicit decompositions Oct 2025 (in OEIS Example section); a(16)=30; keyword 'hard,more'. Caveat: k(N) allows n_1 > N while A192881 fixes n_1 = n; they coincide in all known cases but this equivalence is not proven here — new cells sharpen the finite picture yet cannot settle the limit question (asymptotic side is a wall). Múgica's OEIS a-file has decompositions achieving the published terms." }, { "problem": 301, "oeis": "A390394", "quantity": "A390394(39) — maximum size of A ⊆ {1..39} with no solution 1/a = 1/b_1 + ... + 1/b_k over distinct elements of A; first uncomputed cell (a(1..38) known, a(38) = 29)", "kind": "next_cell", "lower": { "value": "29", "year": 2025 }, "upper": { "value": "30", "year": 2025 }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Headline asymptotic gap (erdosproblems.com/301, edited 2026-01-16): N/2 <= f(N) — take A = (N/2, N] — and f(N) <= (25/28 + o(1))N ~ 0.8929N via an elementary disjoint-blocks argument S_a = {2a,3a,4a,6a,12a} credited to Wouter van Doorn (site-recorded observation, not refereed). Erdős conjectured f(N) = (1/2 + o(1))N. If b_i need not be distinct the threshold drops to N/2 exactly (Cambie–van Doorn, site-recorded). Sequence created Nov 2025 (Raza), 38 terms, keyword 'hard,more', published Python generator; each new exact cell is a record. Cf. #302, #327." }, { "problem": 302, "oeis": "A390395", "quantity": "A390395(732) — maximum size of S ⊆ {1..732} with no solution 1/a = 1/b + 1/c over distinct a,b,c in S; first cell beyond the b-file (a(1..731) known, a(731) = 606)", "kind": "next_cell", "lower": { "value": "606", "year": 2025 }, "upper": { "value": "607", "year": 2025 }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Headline asymptotic gap (erdosproblems.com/302, retrieved 2026-07-18): (5/8 + o(1))N <= f(N) <= (9/10 + o(1))N. Lower: Cambie's construction — odd integers <= N/4 plus all of (N/2, N] (site-recorded). Upper: van Doorn note (PDF in github.com/Woett/Mathematical-shorts, site-linked; not refereed). If sound, the 5/8 lower bound already answers the specific sub-question 'is f(N) = (1/2+o(1))N?' negatively — flag for the curator; both bounds are site-recorded observations. Colouring version is #303, solved by Brown–Rödl 1991. Sequence created Nov 2025 (Raza), no 'hard' keyword — extending the table is routine; the frontier record is the asymptotic constants, which need proofs not witnesses. Cf. #301, #327." }, { "problem": 304, "oeis": "A097849", "quantity": "A097849(106) — max over 1<=k<=106 of the minimal number of unit fractions needed to write k/106 (row-106 maximum of table A097847); finite proxy for N(b)=max_a N(a,b)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "both", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "b-file ends at n=105 (a(105)=4); recent activity by von Brömssen (May 2026) suggests the table is actively extendable. The open problem itself is asymptotic: Erdős 1950 gives N(b) >> loglog b, Vose 1985 gives N(b) << sqrt(log b) — that gap is a wall (erdosproblems.com/304, edited Dec 2025, still lists Vose as best; checked arXiv:2512.22083 (van Doorn–Tang 2025) — it concerns v(k) for decompositions of 1, related via #293 but does not improve N(b)). Extending A097849 feeds the empirical N(b)-vs-loglog(b) curve. Second OEIS ref: A097847." }, { "problem": 313, "oeis": "A054377", "quantity": "Count of known primary pseudoperfect numbers (N>1 with 1/N + sum_{p|N} 1/p = 1); each is the modulus m of a solution to sum 1/p_i = 1 - 1/m, so a new term is a new solution; infinitude is the open question", "kind": "bounded_below_only", "lower": { "value": "10", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Known terms have exactly n distinct prime factors for n<9; new terms come from structured searches (Wang, 'Port Fillings for Primary Pseudoperfect Numbers', arXiv:2605.21518, 2026), not bounded sweeps — the 9th/10th terms are 31 and 38 digits. The two 2026 terms are known terms but not certified as the in-order a(9)/a(10) (they were moved out of the data field), so the ordered next-cell is also open. Infinitude cannot fall to computation." }, { "problem": 322, "oeis": "A025418", "quantity": "A025418(35) — least integer that is a sum of 3 positive cubes in exactly 35 ways (first uncomputed cell; a(36)-a(39) and a(42) are known, so a(35), a(40), a(41) are the open cells)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "b-file ends at a(34)=373248000000; Johnson's comment gives a(36)=496793088000, a(37)=980922617856, a(38)=209584584000, a(39)=644972544000, a(42)=970299000000 (sequence is not monotone, so these do not bound a(35)). The parent problem is asymptotic and a wall: Mahler 1936 disproved Hypothesis K for k=3 with r_3(n) >> n^(1/12) infinitely often; Erdős/Chowla independently proved r_k(n) >> n^(c/loglog n) for all k>=3; whether r_k(n) > n^c infinitely often for k>=4 is fully open (erdosproblems.com/322, edited Sep 2025). Record-count witnesses (integers with record r_3) also live in this table's orbit. Second OEIS ref: A025456." }, { "problem": 327, "oeis": "A384927", "quantity": "A384927(5001) — max size of S in {1..5001} with a+b never dividing ab for distinct a,b in S; monotonicity forces the cell into {3528, 3529} (a(5000)=3528), so it is a single binary decision", "kind": "next_cell", "lower": { "value": "3528", "year": 2025 }, "upper": { "value": "3529", "year": 2025 }, "witness_side": "both", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Conjectured a(n)/n -> 1/2 (odd numbers are an independent set of density 1/2 since a+b even, ab odd). The open content is whether S can be 'substantially more' than the odds, and the a+b | 2ab variant; van Doorn showed density >= 25/28+o(1) forces a bad pair (via #301 discussion). Extending the table sharpens the empirical excess a(n)-ceil(n/2). Site notes 1 claimed proof and 11 comments not yet incorporated (accessed 2026-07-18) — scout should re-check before major effort. Asymptotic question itself is a wall; table cells are machine-decidable." }, { "problem": 334, "oeis": "A062241", "quantity": "A062241(30) — smallest integer >= 2 that is not a sum of two positive 113-smooth numbers (prime(30)=113); first uncomputed cell of the finite proxy for f(n) = best smoothness bound with n = a+b", "kind": "value_gap", "lower": { "value": "> 2570169839", "year": 2010 }, "upper": { "value": "<= 328878692999", "year": 2026 }, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Exact determination of a(30) needs the full sweep from 2570169840 upward (each m usually certifies fast via a small smooth s with m-s smooth), bounded above by 3.29e11 — heavy but bounded. Parent asymptotic problem: Balog 1989 proved f(n) <<_eps n^(4/(9 sqrt e)+eps) (exponent ~0.2695), best verified bound per erdosproblems.com/334 (edited Apr 2026); conjecturally f(n) <= n^o(1). A search hit (Turku NT seminar) suggested a convolution-estimate improvement toward exponent 103/(256 sqrt e) ~ 0.244 — primary source NOT located, treat as unverified lead. Second OEIS ref: A045535 (fully computed to n=50, so upper bounds exist for cells through a(51))." }, { "problem": 340, "oeis": "A080200", "quantity": "Verification depth for '33 is never a difference of Mian-Chowla terms' (33 = A080200(1), the smallest integer not known to lie in A-A for the greedy Sidon sequence A005282)", "kind": "verified_range", "lower": { "value": "100000", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdős-Graham asked about the growth of A (trivially >> N^(1/3), conjectured N^(1/2-eps): wall) and whether A-A has positive density / contains every integer; 22 was confirmed in A-A (a_15 - a_14 = 204-182). If 33 truly appears, a finite computation closes the first cell; nonmembership is not finitely certifiable — hence exact_feasibility unknown. A080200 currently lists 50 conjectured non-differences. Memory, not CPU, is the binding constraint on deeper verification. Second OEIS ref: A005282 (Reichert 100000-term file, May 2026)." }, { "problem": 342, "oeis": "A002858", "quantity": "Computation frontier for the Ulam sequence u(1)=1, u(2)=2: largest X with all Ulam numbers <= X computed, supporting the empirical density ~0.073979 and quasi-period 21.6016 against Ulam's density-0 conjecture", "kind": "verified_range", "lower": { "value": "1000000000000", "year": 2017 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdős/Ulam questions — density 0? infinitely many pairs differing by 2? eventually periodic differences? — are all asymptotic (wall); computation refutes density 0 empirically (density stable ~0.0740). Known structure: clumps with period 21.601584+, ratio to mean gap 13.52 close to 8/5; a(1e9)=13517664323. erdosproblems.com/342 flags claimed partial results in 18 unincorporated comments (accessed 2026-07-18) — scout should re-check before frontier work." }, { "problem": 345, "oeis": "A001661", "quantity": "A001661(9) — largest number not a sum of distinct positive 9th powers", "kind": "bounded_below_only", "lower": { "value": "> 1.55e17", "year": 2023 }, "upper": null, "witness_side": "lower", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Wiener estimates ~200,000 core-months to finish 9th powers. No cheap certificate on either side; excluded from record hunting." }, { "problem": 357, "oeis": "A364132", "quantity": "A364132(23) — least n such that {1..n} contains an increasing 23-term sequence all of whose consecutive-segment sums are distinct (inverse of Erdős-Harzheim f(n); a(22)=52 is the last known term)", "kind": "next_cell", "lower": { "value": "52", "year": 2023 }, "upper": null, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Open content: growth of f(n) — Weisenberg (via #874) gives f(n) >= (2+o(1)) sqrt(n); whether f(n) = o(n) is open; for the non-monotone variant g(n), Hegyvari 1986 proved (1/3+o(1))n <= g(n) and Coppersmith-Phillips (via #867) gives g(n) <= (2/3 - 1/512 + o(1))n — those asymptotics are walls (erdosproblems.com/357, edited Jan 2026). Companion cell: A364153(14) (non-monotone table, 13 terms known, a(13)=18) is an equally cheap SAT target. Erdős also asked whether sum 1/a_i converges for infinite versions." }, { "problem": 359, "oeis": "A002048", "quantity": "Density/growth of MacMahon's segmented numbers a_k (A002048): Andrews conjectures a_k ~ k log k / loglog k; no bounded quantity with a improvable record gap", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known partial results (erdosproblems.com/359, edited Dec 2025, paraphrased): Porubsky 1977 proved a_k < (log k)^eps * k log k/loglog k infinitely often, and limsup A(x)/pi(x) >= 1/log 2; Andrews 1975 studied the conjecture a_k ~ k log k/loglog k. Erdős asked (n=1 case) whether a_k/k -> infinity and a_k/k^(1+c) -> 0 for all c>0 — neither direction is finitely checkable. Empirical growth-curve extension is trivial (open-easy compute, no record semantics); flagged not-gap-shaped so the scout skips it." }, { "problem": 364, "oeis": "A076445", "quantity": "least n such that n-1, n, n+1 are all powerful (Erdos / Mollin-Walsh: conjectured nonexistent)", "kind": "verified_range", "lower": { "value": "7.38e28", "year": 2026 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "abc conjecture implies only finitely many triples; quadruples trivially impossible (one member is 2 mod 4). Chan (2025) and She (2025) rule out special triple shapes (middle a cube with wings p^3x^2 resp. p^2x^3). The movable rung is the verified bound, but raising 7.38e28 needs certified-complete enumeration of gap-2 powerful pairs: Alekseyev's extended A076445 table (33 terms) is explicitly conjectured-complete only (Pell families via A118894/Sentance), so even the range extension is expert-hard. Erdos also conjectured n_{k+2} - n_k > n_k^c for powerful n_k." }, { "problem": 365, "oeis": "A060355", "quantity": "A060355(40) — smallest k > 10^22 with k and k+1 both powerful (next consecutive-powerful pair; the counting function N(x) feeds the open (log x)^{O(1)} question)", "kind": "next_cell", "lower": { "value": "1e22", "year": 2011 }, "upper": { "value": "23146276081390728245000", "year": 2024 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The yes/no part (must one of n, n+1 be a square?) is already answered NO: Golomb 1970 (12167 = 23^3, 12168 = 2^3*3^2*13^2) and Walker 1976 (7^3 x^2 = 3^3 y^2 + 1 has infinitely many solutions). Open content: is N(x) = #pairs <= x bounded by (log x)^{O(1)}? Window contains ~1.1e11 powerful numbers (density 2.173*sqrt(x)); chunked c^2 d^3 enumeration + adjacency scan is hours-scale on one box and would add a new OEIS term (first extension since 2011). The same sweep advances #366's frontier. Of the 39 known pairs only 7 have neither member square (Schoenfield, Jun 2024)." }, { "problem": 366, "oeis": "A060355", "quantity": "count of n with {n, n+1} both powerful and at least one member 3-full (Erdos-Graham: is (8,9) essentially the only one?) — known count = 2 below 10^22", "kind": "verified_range", "lower": { "value": "1e25", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C2", "note": "Both known examples have the 3-full member FIRST (8 = 2^3 before 9 = 3^2; 12167 = 23^3 before 12168), so the exact Lean-stub orientation is even more open. abc implies finitely many such n (Turturean, per erdosproblems). DONE 2026-07-26 (certificates/erdos-366/): swept to 10^25, zero strict-orientation solutions. The lever was enumerating the CUBEFULL side (~X^(1/3)) instead of the powerful-PAIR side (~X^(1/2)) -- three decades for 6.27 core-hours. Extending the A060355 PAIR sweep itself remains the expensive rung and is untouched; constructive Pell-style witnesses look blocked because 3-fullness of a pair member is a much stronger Diophantine constraint. Weaker open sub-question (Er76d): any pair of consecutive 3-full integers at all. Page freshly edited 2026-07-16." }, { "problem": 367, "oeis": "A057521", "quantity": "limsup behavior of prod_{n <= m < n+k} B_2(m) / n^2 for fixed k >= 3, where B_2 = powerful part (A057521) — conjectured << n^{o(1)}", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "van Doorn (per erdosproblems, page edited 2026-03-23): for k <= 2 trivially << n^2, but the clean <<_k n^2 version FAILS for all k >= 3 — prod over [n, n+2] is >> n^2 log n infinitely often. Equivalent up to constants to #935. Empirical sub-lane a scout could run: hunt record values of prod_{m in [n, n+3)} B_2(m)/n^2 via CRT constructions (large squares dividing three consecutive integers) — each example cheaply verifiable but moves no bound. r-full generalization (B_r, r >= 3) also open." }, { "problem": 368, "oeis": "A074399", "quantity": "growth of F(n) = largest prime factor of n(n+1) — conjectured F(n) >> (log n)^2 always, with F(n) < (log n)^{2+eps} infinitely often", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "State of the art verified: Polya 1918 (F -> infinity), Mahler 1935 (>> loglog n), Schinzel 1967-68 (F(n) <= n^{O(1/logloglog n)} i.o.; liminf F(n)/loglog n >= 2/7 effective), and the current best lower bound Pasten 2024 (arXiv:2312.03566, Cor. 1.5): F(n) >> (loglog n)^2 / logloglog n. Empirical sub-lane: Stormer-Lehmer Pell enumeration of p-smooth n(n+1) pairs (largest-n-per-prime table = A002072) supplies certified record smooth pairs; a miner must source the current largest completed smoothness bound before any record claim — not sourced here." }, { "problem": 371, "oeis": "A070089", "quantity": "asymptotic density of {n : P(n) < P(n+1)} (conjectured 1/2); unconditional bracket via lower/upper density", "kind": "value_gap", "lower": { "value": "0.2017", "year": 2025 }, "upper": { "value": "0.7983", "year": 2025 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Nearly theorem-shaped: Teravainen 2018 proved the LOGARITHMIC density is exactly 1/2; Tao-Teravainen 2019: asymptotic density 1/2 at almost all scales; Wang 2021: full conjecture conditional on Elliott-Halberstam for friable integers. Empirics to 10^9: 499992458 terms (~x/2, within 0.5 sigma of binomial p=1/2 model; Greathouse, OEIS). Progression of lower bounds: 0.0099 (Erdos-Pomerance 1978), 0.05544 (de la Breteche-Pomerance-Tenenbaum 2005), 0.1063 (Wang 2017), 0.1356 (Wang 2018), 0.2017 (Lu-Wang). Low value for a machine scout — no witness lane." }, { "problem": 373, "oeis": "A003135", "quantity": "solutions of n! = a_1! a_2! ... a_k! with n-1 > a_1 >= ... >= a_k >= 2; largest known n = 16, Hickerson list conjectured complete", "kind": "verified_range", "lower": { "value": "100000", "year": 2005 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Hickerson list: 9! = 7!3!3!2!, 10! = 7!6! = 7!5!3!, 16! = 14!5!2!. The k=2 case is verified to n <= 10^3000 (Habsieger 2019, after Caldwell 1994) and Erdos 1993 proved a_1 >= n - 5 loglog n (Bhat-Ramachandra: constant 1/log2). Luca 2007: finiteness follows from abc; also would follow from P(n(n+1))/log n -> infinity (#368). The general-k exhaustive frontier is still McCranie's 2005 check at 10^5 — softest verified bound in this chunk; candidates require n, n-1 composite plus a prime-free tail interval, so pushing to ~10^8 is an hours-scale sieve+recursion. A new SOLUTION is believed nonexistent; the honest machine rung is the verified range." }, { "problem": 374, "oeis": "A388851", "quantity": "order of growth of |D_k intersect [n]| for k = 3..6, where F(m) = least k >= 2 admitting a_1 < ... < a_k = m with a_1!...a_k! a perfect square (e.g., is |D_6 intersect [n]| >> n?)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known structure (Erdos-Graham 1976 + 2025-26 OEIS work): D_2 = {squares > 1}; no D_k contains a prime; least element of D_6 = 527; D_k empty for k > 6; |D_3 cap [n]| = o(|D_4 cap [n]|); conjectured |F_3 cap [n]| ~ c sqrt(n) with c = 3.70975... (A389117). Active enumeration lanes (Tao, Wu, Corneth, Pfoertner, Sep 2025-Apr 2026): A388851 (F_3 members, b-file 5685 terms), A387184 (F_4 members, 10000 terms), A389148 (composites e > 6 with no 5-tuple, 10000 terms; first non-semiprime term 16523 = 13*31*41). See Tao arXiv:2603.27990 sect. 1.3. Extending these enumerations is easy machine work but there is no bounded record quantity for the stated problem." }, { "problem": 376, "oeis": "A030979", "quantity": "A030979(1375) — least k > 10^70 with binomial(2k,k) coprime to 105 (equivalently: all base-3 digits <= 1, base-5 digits <= 2, base-7 digits <= 3)", "kind": "next_cell", "lower": { "value": "1e70", "year": 2015 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Graham's $1000 prize question is INFINITUDE of the sequence — that side is a wall (EGRS 1975 settles any two odd primes; three primes open; Bloom-Croot [BlCr25]: almost-all-digits version for large primes). Pomerance heuristic: ~x^0.02595 terms up to x, so new terms beyond 10^70 exist but are extremely sparse; naive digit-DFS explodes (2^147 base-3 strings to 10^70) — needs Thompson-style multi-base constraint propagation, hence plausible-not-easy, and exact-cell status beyond 10^70 is unproven at realistic scale. b-file unchanged as of OEIS rev 2026-06-28. Also note Graham expected 3160 to be the last term coprime to 3*5*7*11." }, { "problem": 382, "oeis": "A388850", "quantity": "A388850(9) — least start u of a maximal run of exactly 10 consecutive integers whose product has its largest prime factor appearing with exponent >= 2 ('bad interval' of width 9)", "kind": "next_cell", "lower": null, "upper": { "value": "4928180396", "year": 2026 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "a(0..8) = 4, 8, 48, 1680, 76725, 332925, 7474752, 4541154, 75047565 known exact (von Bromssen, Sep 2025; note non-monotone, so no clean lower bracket published — exhaustive frontier is at least ~7.5e7 implied by exactness of a(8), exact frontier unstated). Open candidates: a(9) <= 4928180396, a(10) <= 40533366231, a(11) <= 111460496439, a(12) <= 436502026483 (Sungkawichai, Jan 2026), a(13) <= 17952448643078 (StijnC, per OEIS). A segmented largest-prime-factor sieve to 4.93e9 + sliding-window check settles a(9) in hours on one box, and the same infrastructure attacks a(10)-a(12). Each newly confirmed larger width is also a direct witness for Erdos's 'can v-u be arbitrarily large?' (Cambie heuristic: yes; Cramer would force v-u = v^{o(1)}; Ramachandra gives v-u <= v^{1/2+o(1)}). Best actionable cell in this chunk alongside #365." }, { "problem": 385, "oeis": "A322292", "quantity": "Largest member of A322293 = {k : A322292(k) <= k}, where A322292(k) = max over composites c < k of (c + lpf(c)) — the exceptional k where no composite c < k has c + lpf(c) > k; the set is conjectured finite (Erdős–Eggleton–Selfridge)", "kind": "bounded_below_only", "lower": { "value": "267680", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "A322292(k) is trivially <= k + sqrt(k); Erdős–Eggleton–Selfridge suggest exceptional k are finite and that the max may be >= k + (1−o(1))·sqrt(k) [Er79d p.73; ErGr80 p.74]. A322293 has exactly 100 known terms, complete below 10^8; largest 267680 (pair 267672/267680 after a long gap from 8742 — thinning consistent with finiteness). Bloom's page: equivalent to problem #430 (Adenwalla observation); Tao has a blog discussion; Tao marks it difficult. Record = any new exceptional k in (10^8, ∞): a segmented lpf-sieve scan 10^8 → 10^11 is GPU-hours (candidate k must have k−1 prime, which prunes hard), but further terms may simply not exist — graded plausible, not open-easy. Exact resolution (finiteness) is a wall." }, { "problem": 386, "oeis": "A280992", "quantity": "A280992(8) — the next squarefree triangular number that is a product of consecutive primes (equivalently the next n > 715 with C(n,2) a product of consecutive primes); known terms 1, 3, 6, 15, 105, 210, 255255", "kind": "next_cell", "lower": { "value": "> 12500002500000", "year": 2017 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Known n with C(n,2) such a product: 4, 6, 15, 21, 715 (terms 6, 15, 105, 210, 255255). Erdős–Graham [ErGr80 p.74] wrote that proving finiteness even for C(n,2) seems hopeless; their guess that it never happens for 3<=k<=n−3 was refuted by Weisenberg: C(7,3)=5·7, C(10,4)=2·3·5·7, C(14,4)=7·11·13, C(15,6)=5·7·11·13. Cheap record available: the exhaustive floor 1.25e13 can be pushed to ~1e24 with only ~4e10 square tests (prime pairs p·p'' with p <= 1e12 dominate; longer windows are far fewer) — GPU-hours for an 11-order-of-magnitude lower-bound record on a(8). But heuristic solution count grows only like log log X (pair case: sum of 1/sqrt(8·p·p'') over consecutive prime pairs), so an actual 8th term is unlikely in reach — witness graded hard. a(8) may not exist at all → exact unknown." }, { "problem": 389, "oeis": "A375071", "quantity": "A375071(27) — least k >= 1 such that 27·28···(26+k) divides (27+k)···(26+2k) (first uncomputed cell; a(0)–a(26) known, a(26) = 5048891644620)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "record_in_progress", "confidence": "C3", "note": "Erdős–Straus: does such k exist for every n? (Open; existence for each computed n is witnessed.) Mehta computed n <= 18; OEIS extended to a(26) with active work Mar 2026 (Kesarwani comment; SharkyKesa listed as currently working on erdosproblems.com — coordinate, don't race). The (n,k) → (n+k−1, n) transform ties this to A375077/problem 396: transform preserves validity but NOT minimality (a(7) = 8171 is much smaller than the transform predicts), so the candidate a(27) above is an upper-bound witness at best and must be verified before recording. Exhaustive minimality sweeps at the ~1e13 scale are demonstrated feasible by the A375077 project → exact cell." }, { "problem": 390, "oeis": "A193429", "quantity": "Asymptotic constant c in f(n) − 2n ~ c·n/log n, where f(n) = A193429(n) = minimum possible largest factor when n! is written as a product of integers all > n", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Guy–Selfridge [EGS82] proved f(n) − 2n ≍ n/log n (both order bounds); the open question is whether f(n) − 2n ~ c·n/log n for a specific constant c. The stub's Lean formalization marks 'research solved' — that plausibly reflects the proven ≍ statement; the constant question is the open residue per erdosproblems.com. a(n) = 0 only for n = 1, 2 (Adams-Watters); sequence appears O(n). Empirical c-estimation from an extended table (b-file n <= 168, Kesarwani) is possible scout work but certifies nothing → not gap-shaped." }, { "problem": 393, "oeis": "A388302", "quantity": "A388302(80) — least spread b_t − b_1 over factorizations 80! = b_1···b_t into t >= 2 distinct increasing integer factors (first uncertified cell; table certified for n = 2..79 with witness tuples)", "kind": "next_cell", "lower": null, "upper": { "value": "78", "year": 2025 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "f(n) = A388302(n). Headline open question: is f(n) = 1 infinitely often — f(n)=1 ⟺ n! = m(m+1) ⟺ 4·n!+1 is a perfect square; known only n = 2, 3 (Brocard-adjacent). Berend–Osgood 1992: #{n<=N : f(n)=m} = o(N) for each fixed m; Bui–Pratt–Zaharescu 2023: ≪_m N^{33/34}; Luca 2002: ABC ⇒ f(n) → ∞. Existing table n = 2..79 by Corneth (PARI program + tuple list linked from OEIS); a(n) <= min(A061057(n), n−2). Record: extend the certified table from a(80) onward — candidate discovery via the geometric-mean window heuristic is hours; exhaustive minimality per cell is the honest certification bar. Entry created Sep 2025; no newer bounds found." }, { "problem": 394, "oeis": "A344005", "quantity": "Asymptotics of Sum_{n<=x} t_2(n), where t_2(n) = A344005(n) = least m with n | m(m+1)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Hall 1978 [ErHa78] proved Sum_{n<=x} t_2(n) ≪ x²·logloglog x/loglog x, settling Erdős's o(x²) conjecture; trivial lower bound ≫ x²/log x from t_2(p) = p−1; Erdős–Hall conjecture o(x²/(log x)^c) for every c < log 2 — that conjecture is the open gap and is not witness-movable. Structural facts: t_2(n) <= n−1 with equality iff n is a prime power; <= n/2 − 1 for non-prime-powers; >= (sqrt(4n+1)−1)/2 (Sloane 2021). Secondary discrete asks on the page — behaviour of t_{n−3}(n!), and whether infinitely many n satisfy t_k(n!) < t_{k−1}(n!) − 1 for all 1 <= k < n (known only for n = 10, Erdős–Hall–Selfridge) — could seed a future bounded-search lane (find a second such n) but are not the headline bounded quantity, so this entry stays not_gap_shaped." }, { "problem": 396, "oeis": "A375077", "quantity": "A375077(16) — least k such that k(k−1)···(k−16) divides C(2k,k) (first uncomputed cell; a(1)–a(15) listed, a(15) = 2394789405254721)", "kind": "next_cell", "lower": { "value": ">= 18253129921842", "year": 2026 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "record_in_progress", "confidence": "C3", "note": "Flagship machine-shaped cell in this chunk, but ACTIVELY OWNED: minimality of a(8)–a(13) is certified by exhaustive search plus a small-prime sieve whose completeness ('Small Prime Barrier Theorem') is formally verified in Lean 4 + Mathlib (Justin Dehorty, Mar 2026; repo github.com/jdehorty/erdos396 with validation certificates — linked from OEIS, not independently audited here). Certification status of listed a(14), a(15) is not stated in the OEIS comment; presuming them minimal would lift the floor for a(16) to >= 2394789405254721. Expected a(16) ~ 1e16–1e17 by growth pattern → days-to-weeks with the published optimizations. Dehorty and SharkyKesa are listed as currently working on erdosproblems.com/396 — coordinate rather than race; certifying a(14)/a(15) minimality is a complementary sub-record. Solutions transform to problem 389 upper bounds via (n,k) → (n+k−1, n). Context: n+1 | C(2n,n) always (Catalan); Pomerance [Po14] also shows the set of n with n | C(2n,n) has upper density < 1/3." }, { "problem": 398, "oeis": "A146968", "quantity": "Least n > 7 with n! + 1 a perfect square (Brocard–Ramanujan; conjecturally none — known solutions n = 4, 5, 7 with m = 5, 11, 71)", "kind": "verified_range", "lower": { "value": "10^15", "year": 2020 }, "upper": null, "witness_side": "upper", "witness_feasibility": "historic", "status": "open", "confidence": "C3", "note": "Open since 1876 (Brocard) / 1913 (Ramanujan). Overholt 1993: weak ABC ⇒ finitely many solutions. Naciri 2025 [Na25, per erdosproblems.com]: finiteness if m±1 is k-free (any k >= 2) or a prime power; if 7-free then n = 4, 5, 7 are the only solutions. A085692: the next square, if any, exceeds 10^20000 (Wasserman 2005) and a(4) > (10^12+1)! per Matson's claimed range. FRESHNESS FLAG: the 10^12 and 10^15 tiers are unrefereed code-based claims (Matson PDF; Epstein–Glickman repo) — the safest citable floor is Berndt–Galway 10^9; recorded value uses the strongest claim with provenance flagged. Range extension past 10^15 via GPU QR-sieving is a plausible lane; a 4th solution would be historic. Secondary gap on this problem's other OEIS ref A141399 (k whose k(k+1) prime support is a consecutive run from 2): 28 terms, largest 633555, complete to 5·10^8 (Chandler 2009), and a(29) > 2.29·10^25 if it exists (Resta 2019)." }, { "problem": 408, "oeis": "A049108", "quantity": "Growth/normal-order constant of f(n) = A049108(n), the number of iterations of Euler phi needed to send n to 1 (does f(n)/log n converge, etc.)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Pillai 1929 [Pi29]: log_3 n < f(n) < log_2 n for all large n; Shapiro 1950 [Sh50]: f is essentially multiplicative. Erdős–Granville–Pomerance–Spiro 1990 [EGPS90] answered the problem's first two questions affirmatively CONDITIONAL on a form of the Elliott–Halberstam conjecture — the unconditional statement is the wall. Also expected: for almost all n the largest prime factor of the k-th phi-iterate is n^{o(1)} whenever k → ∞. Discussed in Guy B41. A049108 = A003434 + 1. No bounded machine record identified." }, { "problem": 409, "oeis": "A229487", "quantity": "A229487(n), n = 1..53 — per-cell greatest integer whose iteration x → phi(x)+1 reaches prime(n); every listed maximum is conjectural (max-type record surface for Finucane's problem)", "kind": "bounded_below_only", "lower": { "value": "per-cell table of 53 conjectured maxima (largest listed: a(53) = 54582 for p = 241; e.g. a(6) = 138 for p = 13, a(1) = 2, a(2) = 6)", "year": 2013 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Finucane's problem: every trajectory of x → phi(x)+1 reaches a prime (trivially, by descent); A039651 counts the steps; Cambie notes F(n) = 1 infinitely often and F(n) = o(n) is trivial — Bloom reads the intended open content as good upper bounds for the iteration count F(n). Two machine lanes on the A229487 surface: (1) forward phi-sieve over all x <= 1e9–1e10 (hours) rederives every basin and surfaces any converger beating a listed cell — Noe's original search bound is unstated, so even a clean replication with an explicit bound upgrades the table's provenance; (2) per-cell CERTIFICATION: enumerate the full inverse-image tree of prime(n) under x → phi(x)+1 (preimages of y are the inverse-totient set of y−1, finite and computable since phi(x) >= sqrt(x/2) forces x <= 2(y−1)²); if the recursion terminates, the conjectured maximum becomes exact — converting a 'conjectured' OEIS cell into a theorem is a legitimate record → exact cell. The page also poses the analogous sigma(n)−1 iteration (does it always reach a prime?), which has no OEIS record surface here." }, { "problem": 410, "oeis": "A007497", "quantity": "A007497(1501) — next term of the iterated sum-of-divisors trajectory 2, σ(2), σ(σ(2)), … underlying the question whether lim_k σ_k(n)^(1/k) = ∞", "kind": "next_cell", "lower": { "value": "1500 terms known; a(1500) has 1393 digits", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Underlying problem (Erdős–Granville–Pomerance–Spiro 1990, problem (iii) p.169; Guy B9) is explicitly not resolvable by finite computation — the trajectory extension is the machine-record lane, feeding growth heuristics (Cloitre's conjectured (1/2)log n < a(n+1)/a(n) < 2 log n; divisibility patterns checked to n=1000 per OEIS). Factor-chain propagation (σ splits over prime powers of the previous term) keeps most steps in ECM range, but a single hard 150+ digit cofactor (e.g. p+1 for a large prime factor p) can stall the chain — hence plausible, not open-easy. Cohen–te Riele (Exp. Math. 5, 1996) is the classical computational study." }, { "problem": 411, "oeis": "A383044", "quantity": "Smallest solution n of φ(n)+φ(n+φ(n)) = n whose odd part is outside {1,3,5,7,35,47} — equivalently the smallest exceptional prime p ≡ 7 (mod 8) with φ((3p-1)/4) = (p+1)/2 (the r=2 case of Erdős–Graham's g_{k+r}(n) = 2 g_k(n) for g(n) = n+φ(n))", "kind": "bounded_below_only", "lower": { "value": "1e10 (no exceptional prime below this)", "year": 2025 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Min-type quantity: a found exceptional prime creates the upper bound and yields a brand-new solution family (disproving Cambie's conjecture that none exist). Heuristically the exceptional set may be empty — tied to whether φ(m) = 2(m+1)/3 has infinitely many solutions — so the cheap-witness hunt is graded hard. The complementary record (pushing the exclusion bound 1e10 → 1e12 with a segmented φ sieve over numbers ≡ 5 mod 6) is a days-scale job but is auditable only by recomputation, not by a succinct certificate. Known solution families (per erdosproblems.com/411, paraphrased): odd part in {1,3,5,7,35,47} times powers of 2; base r=2 solutions n=10 and 94; Weintraub found period-9 and a period-25 identity at n=3114 with ratio 729; Cambie found further sporadic period-4 identities. A383044 b-file has 123 terms; De Vlieger verified the family structure up to 2^24." }, { "problem": 412, "oeis": "A051572", "quantity": "A051572(524) — next term of the iterated-σ trajectory of seed 5; the frontier of the (m,n) = (2,5) instance of the van Wijngaarden/Erdős merging conjecture (∃ i,j with σ_i(m) = σ_j(n))", "kind": "next_cell", "lower": { "value": "523 terms known; a(523) has 437 digits; no common value with the 1500 known iterates of seed 2 (intersection of b-files checked empty 2026-07-18)", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdős attributed the conjecture to van Wijngaarden (1950s); Erdős–Graham thought nothing provable soon; Selfridge reported numerical evidence suggesting the answer may be NO — i.e. the merge witness may not exist, so the durable machine record is extending both trajectories (A051572 at 523 terms is far behind A007497 at 1500) and re-running the collision check. Bootstrap re-factoring of the 523-step chain from seed 5 is days of ECM work with the usual hard-cofactor risk. Cross-refs: A007497 (seed 2), A257348. See also problems 413 and 414." }, { "problem": 413, "oeis": "A005236", "quantity": "A005236(10001) — next barrier for ω (an n with m + ω(m) ≤ n for all m < n); current frontier a(10000) = 2054598", "kind": "next_cell", "lower": { "value": "10000 barriers known; a(10000) = 2054598", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Infinitude of ω-barriers is OPEN (Erdős believed yes; Guy B8; Erdős–Graham noted sieve methods not yet strong enough). Recent movement: per erdosproblems.com/413 remarks (page edited 2026-04-17), Lau (2026, arXiv:2604.15042) answered the ε-relaxation (∃ε>0 with infinitely many n s.t. m + εω(m) ≤ n for all m < n) positively and proved a weakened main statement with the condition restricted to m ≤ n − C; we checked only the abstract, not the theorem statements — treat as reported. Erdős (1979) proved barriers for F(n) = Π k_i (product of exponents) have positive density; Selfridge: largest Ω-barrier below 1e5 is 99840. Related: Tao–Teräväinen arXiv:2512.01739." }, { "problem": 416, "oeis": "A264810", "quantity": "Whether V(2x)/V(x) → 2 and whether an asymptotic formula exists for V(x) = #{n ≤ x : φ(m) = n solvable} (the totient value-counting function)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Order of magnitude essentially settled by proof, not computation: Pillai (1929) V(x) = o(x); Erdős (1935) V(x) = x(log x)^(-1+o(1)); Maier–Pomerance (1988) V(x) = (x/log x)·e^((C+o(1))(logloglog x)^2); Ford (1998) sharp two-sided ≍ with explicit constants C1, C2, C3 — just short of an asymptotic formula, which is exactly what remains open. Guy B36. No witness lane: excluded from the records hunt." }, { "problem": 417, "oeis": "A061070", "quantity": "lim_{x→∞} V(x)/V'(x), where V(x) = #{totient values ≤ x} and V'(x) = #{φ(m) : 1 ≤ m ≤ x} — does the limit exist, and is it > 1?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Trivially V'(x) ≤ V(x). Erdős (1998) suggested the limit may even be infinite (per erdosproblems.com/417). The formal-conjectures Lean file inverts the ratio to V'(x)/V(x) for boundedness — a technical choice only. Ford's totient machinery (Fo98, see problem 416) is the relevant proof tool; Tao arXiv:2309.02325 is linked from A061070. No witness lane: excluded from the records hunt." }, { "problem": 421, "oeis": "A389544", "quantity": "d*(421) = supremum of asymptotic densities achievable by sequences 1 ≤ d_1 < d_2 < … with all contiguous products Π_{u≤i≤v} d_i distinct (question: is d* = 1?)", "kind": "value_gap", "lower": { "value": "1/e - ε achievable for every ε > 0 (≈ 0.36788)", "year": null }, "upper": { "value": "1 (trivial ceiling; the conjecture is that it is attained)", "year": null }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "HOT: erdosproblems.com/421 shows a claimed solution posted in its comments (unreviewed as of 2026-07-18, 43 comments, several users including Bloom and Alexeev mark it tractable) — the scout should re-check upstream status before investing. Greedy sequence A389544 (Sungkawichai, Jan 2026) has 20000 terms with a(20000) = 20163, prefix density ≈ 0.9919 and rising (complement A390848: 1, 6, 12, 16, …), consistent with the conjectured answer yes. Source: ErGr80 p.84." }, { "problem": 422, "oeis": "A005185", "quantity": "W(422) = largest N such that Hofstadter's Q-sequence f (f(1)=f(2)=1, f(n)=f(n-f(n-1))+f(n-f(n-2))) is verified well-defined (both back-indices ≥ 1) for all n ≤ N", "kind": "verified_range", "lower": { "value": "3e10", "year": 2023 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "The Erdős–Graham/Hofstadter questions — does f miss infinitely many integers, and what is its behavior — are open and explicitly not machine-closable; it is not even known that f is total, which is exactly what the verified range tracks. Advancing 3e10 → 1e11+ is a memory-engineering job (mmap/disk streaming, days), a clean citable record with a 2014→2023 precedent chain in OEIS. Context literature: Pinn (1999/2000) order-and-chaos analyses; Deane–Gentile arXiv:2311.13854 (diluted variant with existence proofs); Guy E31." }, { "problem": 423, "oeis": "A005243", "quantity": "Growth exponent θ = inf{t : a_n ≪ n^(t+o(1))} for the Hofstadter consecutive-sum sequence (a_1=1, a_2=2, a_k = least integer > a_{k-1} expressible as a sum of ≥ 2 consecutive earlier terms)", "kind": "value_gap", "lower": { "value": "1 (a_n ≥ n trivially; refined lower bound a_n = n + Ω(log log n))", "year": 2026 }, "upper": { "value": "688/413 + o(1) ≈ 1.6659 (as stated on erdosproblems.com via Cushman's convex-set sumset bound; Tang's own abstract states 4175/2506 + o(1) ≈ 1.6660)", "year": 2026 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "MAJOR 2026 movement: the original omission question is now CLOSED — Bolan and Tang independently proved infinitely many integers never appear (a_n − n is nondecreasing and unbounded), per erdosproblems.com/423 (edited 2026-03-23) and Tang arXiv:2603.09939. The hub stub still tags the whole problem open: the asymptotics half genuinely remains open (truth likely a_n = n + o(n)), but the solved half should be recorded on the next hub refresh. Asked by Hofstadter (Erdős says inspired by Ulam); Guy E31; b-file 1000 terms, a(1000) = 1149." }, { "problem": 424, "oeis": "A005244", "quantity": "Lower asymptotic density of the Hofstadter product-minus-one closure {2,3} -> a_i*a_j - 1 (A005244); conjectured positive", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "erdosproblems.com marks this 'cannot be resolved with a finite computation'. Known structure: no integer == 1 (mod 3) ever appears (residues stay in {0,2} mod 3), so upper density <= 2/3; the original 'almost all integers' phrasing in ErGr80/Guy E31 is trivially false and the intended question (Er77c) is positive lower density. OEIS keyword 'nice,easy' refers to term generation, not the problem. A possible scout instrument (non-record): compute counting function c(x)/x to large x from the closure BFS to watch for density stabilization, but this cannot certify anything." }, { "problem": 436, "oeis": "A000445", "quantity": "A000445(8) = Lambda(8,2) — latest possible first occurrence (limsup over primes p) of a pair of consecutive 8th-power residues mod p; first uncomputed cell of the Lehmer-Lehmer table", "kind": "next_cell", "lower": { "value": "1499876", "year": 2019 }, "upper": { "value": "1508324", "year": 2019 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Gap of only 8449 between Reble's bounds has stood since 2019/2020 (OEIS keyword 'more,hard'; freshness re-checked via web search 2026-07-18, nothing newer found). Values Lambda(k,2) for k=2..7: 9, 77, 1224, 7888, 202124, 1649375 (Lehmer-Lehmer 1962 / Bierstedt-Mills 1963 / Dunton 1965 / LLM 1963 / BLL 1964, per OEIS). Note a(8) < a(7). Hildebrand 1991 proved Lambda(k,2) finite for ALL k, so every cell k >= 8 is a legitimate finite target. Remaining fully-open content of #436: is Lambda(k,3) finite for odd k >= 5 (known: Lambda(3,3)=23532, LLMS 1962; = infinity for even k and for all m >= 4 by Lehmer-Lehmer + Graham 1964), plus growth rates. Exact determination of a(8) closed the same way as k<=7 is a bounded (if large) computation — good expert-tooling target." }, { "problem": 451, "oeis": "A386620", "quantity": "A386620(184) = n_184 — smallest k > 368 such that (k-1)(k-2)...(k-184) has no prime factor in (184, 368); first uncomputed cell (b-file complete through n=183)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "The open PROBLEM is the asymptotic estimate of n_k, which is a wall for machines: van Doorn-Tang (arXiv:2606.19863, June 2026, in OEIS formula section) prove n_k > exp(c (log k)^2/loglog k); trivial upper n_k < e^{O(k)} (PNT / product of primes in (k,2k), Adenwalla observation on erdosproblems.com); Erdos conjectured k^d-beating growth but e^{o(k)}. The RECORD lane is table extension: b-file has exactly 183 terms (Kalogeropoulos 1-70, Chai Wah Wu 71-92, Greathouse 93-183). Because each candidate k is admissible iff k avoids 184 residues mod each of 31 primes, admissible density is prod (p-184)/p over p in (191..367) ~ tiny — terms jump erratically (a(36)=579124 vs a(183)~1.7e17), so budget per new cell is uncertain; grade honest: days of sieve, not hours." }, { "problem": 454, "oeis": "A389677", "quantity": "limsup_n (f(n) - 2*prime(n)) where f(n) = min_{0= 1; record = largest X with the inequality verified for all k with p_{k+1} <= X", "kind": "verified_range", "lower": { "value": "1e12", "year": 2026 }, "upper": null, "witness_side": "both", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "erdosproblems.com marks this FALSIFIABLE (single-k counterexample possible) and 'almost certainly true but proof beyond reach': ruling out two primes-squared in one prime gap is Legendre-conjecture strength. A counterexample needs a prime-free interval of length ~2*sqrt(p_k) — astronomically beyond all known gaps, so the falsification lane is dead; the honest record is the verified range. Cheap extension route to 4e18: all prime gaps below 4e18 are < 1476 (Oliveira e Silva-Herzog-Pardi maximal-gap verification, 2014), which kills double-square cohabitation for q > 737, leaving only a small square-cube/higher coincidence enumeration — an afternoon job for the rungs fleet. No published verified range was found in OEIS or on the problem page; this 1e12 record is ours." }, { "problem": 462, "oeis": "A032742", "quantity": "Existence of C > 0 with sum of p(n)/n over n in [x, x + C*x^(1/2)*(log x)^2] bounded below by an absolute constant for all large x (p(n) = least prime factor)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Background fact stated on the problem page: sum over composite n < x of p(n)/n is asymptotically c*x^(1/2)/(log x)^2, so the conjectured interval length matches the mean spacing of the sum's mass (dominated by n = 2*prime). The linked OEIS sequence A032742 (largest proper divisor, i.e. n/p(n)) is auxiliary only — no bounds live there. No partial results found on the problem page (no remarks section beyond the statement) or in OEIS. Not a records-lane target." }, { "problem": 468, "oeis": "A167485", "quantity": "Sub-conjecture record for the f(N)=o(N) question: A167485(N) > 0 (every N is an initial-segment divisor sum of some m) for all N except 2 and 5 — verified through N = 20000", "kind": "verified_range", "lower": { "value": "20000", "year": 2024 }, "upper": null, "witness_side": "both", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The actual Erdos question — is f(N) = o(N), where f(N) = A167485(N+1)-indexed minimal m — is asymptotic (wall). Data points: liminf a(n)/n = 0 is PROVEN (Israel: a(sigma(n)) <= n and sigma(n)/n unbounded, OEIS comment 2024); but the ratio spikes — max a(n)/n = 11.84 at n=17567 within the b-file (computed this session), so o(N) is far from empirically settled. Companion sequences (Kovac, Aug 2025): A387502 = number of debut sums per n (record indices 2,12,36,144,336,1320,1980,5040,8400,25200,75600 per OEIS comment; average ~0.83 over first 1e5), A387503 = its partial sums; both b-files to 10000. The |D_n \\ union D_m| part of #468 is served by A387502 but has no bound-shaped open gap recorded anywhere we found." }, { "problem": 469, "oeis": "A006036", "quantity": "Convergence of sum of 1/n over primitive pseudoperfect numbers (A006036); companion question for A119425 primitives", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Origin: Benkoski-Erdos 1974 (Math. Comp. 28, 617-623) and Erdos 1970; also Guy B2. Odd terms exist (945 is the first odd primitive pseudoperfect, in OEIS data; odd terms coincide with odd primitive abundant A006038 as far as computed, Kehowski 2005 comment). FRESHNESS FLAG: erdosproblems.com/468-neighbor page for 469 shows '1 claimed proof for this problem' in its comment metadata (page last edited 2025-10-19, status still OPEN as of 2026-07-18) — the claim is unvetted; the integration lane should re-check the page before treating this entry as stable. Counting-function growth of A006036 vs A119425 partial sums is a possible instrument but certifies nothing." }, { "problem": 470, "oeis": "A006037", "quantity": "Existence of an odd weird number (sigma(n) >= 2n, n not a sum of any subset of its divisors) — $10 Erdos prize; record = exhaustive search bound", "kind": "verified_range", "lower": { "value": "1e21", "year": 2014 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Second half of #470 — are there infinitely many primitive weird numbers (A002975, b-file 1161 terms)? — is proven CONDITIONALLY by Melfi 2015 (J. Number Theory 147, 508-514) under a prime-gap hypothesis p_{n+1}-p_n < 0.1*sqrt(p_n); unconditional infinitude open; Amato-Hasler-Melfi-Parton 2019 construct primitive weirds with many prime factors. Benkoski-Erdos 1974 proved weird numbers have positive density; if no odd weird exists, every weird number has abundancy < 4 (page remark, cross-ref problem 825). Feasibility graded hard, not none: a multi-year BOINC project exhausted 1e21 and the frontier heuristics (Fang: near-perfect abundance boundary) suggest any odd weird is enormous. Freshness re-checked by web search 2026-07-18: 1e21/1e28 still the cited records." }, { "problem": 472, "oeis": "A389713", "quantity": "Survival length of the Ulam prime chain seeded (3,5): q_{n+1} = smallest prime of the form q_n + q_i - 1; verified alive through 10000 terms (a(10000) = 190669)", "kind": "verified_range", "lower": { "value": "10000", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Problem due to Ulam (via Erdos-Graham 1980): does SOME finite prime seed extend to an infinite chain? The (3,5) chain (which begins 3,5,7,11,13,...) is the canonical candidate and 'possibly infinite' per the problem page; OEIS comment confirms infinitude unknown. Heuristically the chain has ~n chances per step each prime with probability ~1/log(q_n), so death probability decays rapidly — expect verification to extend essentially forever without deciding anything; the interesting machine event would be discovering a seed whose chain dies (none recorded anywhere we found) or structure enabling a proof. b-file extension beyond 10000 is a free OEIS contribution but low scientific value; grade the lane accordingly despite the open-easy verifier." }, { "problem": 478, "oeis": "A210184", "quantity": "Socialist-prime sub-question of #478: does any prime p>5 have 2!,3!,...,(p-1)! all distinct mod p (equivalently A210184(n)=prime(n)-2)? The headline conjecture |A_p| ~ (1-1/e)p is asymptotic-only.", "kind": "verified_range", "lower": { "value": "2^40 = 1099511627776", "year": 2019 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "CAUTION: Abramov arXiv:2604.26429 (submitted 2026-04-29, 6 revisions through 2026-07-16, unrefereed) CLAIMS an elementary proof that no socialist prime > 5 exists — if it holds, this witness lane is dead; erdosproblems.com/478 (edited 2026-04-12) still lists the problem open with 0 claimed proofs. Headline: best lower |A_p| >= (sqrt(2)-o(1))sqrt(p) (Grebennikov–Sagdeev–Semchankau–Vasilevskii, GSSV24); upper p-2 (Wilson) — asymptotic, no finite witness moves it. Heuristically socialist primes should be essentially nonexistent (injectivity of a pseudo-random map has probability ~e^{-cp})." }, { "problem": 479, "oeis": "A036236", "quantity": "A036236(1119) — least n with 2^n ≡ 1119 (mod n); first of 93 unknown entries among k <= 10^4 (Graham's conjecture: for every k ≠ 1 there are infinitely many such n)", "kind": "next_cell", "lower": { "value": "> 1e13", "year": 2024 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Known terms can be astronomically larger than k (a(3)=4700063497). Structured CRT/multiplicative-order constructions (Crump/Alekseyev style) can leapfrog the sieve — the 93 residues survived structured searches to 10^13, so not open-easy. A witness gives only an upper bound on the least n; exactness requires closing the exhaustive scan below it (same compute class as the existing 10^13 sieve if the witness lands near it, unbounded in principle — hence exact=unknown). k=1 provably impossible (Alekseyev proof in OEIS). Graham conjecture proved for k=2^i, i>=1, and k=-1 (per erdosproblems page; Tang note). Wilson b-file terms 0..1026 complete. Stub lists six companion OEIS ids (A015919 etc.) = solution sets for individual k; A036236 chosen as the frontier carrier." }, { "problem": 483, "oeis": "A030126", "quantity": "A030126(6) — 6-color Schur number in the version-1 convention (least N such that every 6-coloring of {1..N} has a monochromatic x+y=z, x=y allowed); equals S(6)+1 in the largest-N convention", "kind": "next_cell", "lower": { "value": "537", "year": 2000 }, "upper": { "value": "1837", "year": 2004 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "CONVENTION WARNING: A030126 is version 1 = S(k)+1 — do not mix with the largest-N convention (cf. the #552 wheel-convention retraction lesson). S(6) >= 536 unmoved since 2000 despite SAT-era template methods; recent movement is at k>=7: S(7) >= 1740 (arXiv:2107.03560, 2021 — makes the OEIS comment a(7) >= 1681 stale, now a(7) >= 1741) and shifted S-templates with recurrence S(k+2) >= 10S(k)+2 improving S(8), S(13) (arXiv:2607.15034, Jul 2026). Exact closure of a(6) is far beyond current SAT: S(5)=160 closure took Heule's 2-petabyte DRAT (2017). Headline #483 (is f(k) < c^k?) is asymptotic; large-k bounds 380^{k/5}-O(1) <= f(k) <= (e-1/6)k! (Ageron et al. arXiv:2112.03175; Xu–Xie–Chen 2002)." }, { "problem": 500, "oeis": "A140462", "quantity": "Turán density pi(K_4^3) = lim ex_3(n,K_4^3)/C(n,3) — Turán's (3,4)-problem, $500 Erdős prize; finite carrier ex_3(n,K_4^3) vs the conjectured-exact construction value A140462(n)", "kind": "value_gap", "lower": { "value": "5/9 ≈ 0.55556 (density; finite form A140462(n), exact closed formula)", "year": 1961 }, "upper": { "value": "0.561666 (density)", "year": 2010 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "A140462 is a closed-form conjectured construction value (b-file to n=10000), NOT computed extremal data — there is nothing to 'extend'. Frohmader's Omega(6^{n/3}) non-isomorphic conjectured-extremal families make DRAT-UNSAT confirmation blow up past tiny n; the live upper-bound frontier is SDP/flag-algebra machinery, not witness search. Deep audit (2026-07-13) found no movement 2023–2026 beyond l2-norm variants (Balogh et al.). Hub status 'wall'; witness lane exists formally (cheap verifier) but success = disproving a 65-year conjecture." }, { "problem": 503, "oeis": "A175769", "quantity": "A175769(9) — maximum isosceles set in R^9 (every 3 points form an isosceles triangle); first unknown dimension of the Erdős–Kelly problem", "kind": "next_cell", "lower": { "value": "46", "year": null }, "upper": { "value": "55", "year": 1984 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "a(1..8) = 3,6,8,11,17,28,30,45: d=2 Erdős–Kelly 1947 (new proof Kovács 2024 per site), d=3 Croft 1962, d=4,5 Kido, d=6,7,8 Ionin (EJC 16 (2009) R141). Blokhuis bound sharp at d=1,2,6,8 (a(8)=45=C(10,2)) but a(7)=30<36 shows it is not always tight — so a(9) in [46,55] is genuinely open at both ends. Natural candidate witnesses: large two-distance sets (any two-distance set is isosceles by pigeonhole) from association schemes / strongly regular graphs; that classification field (Musin, Nozaki, Lisoněk) is expert-worked, hence 'hard'. Exact determination is a continuous-space classification problem (Ionin-style), not bounded exhaustive search." }, { "problem": 528, "oeis": "A387897", "quantity": "C_2 — connective constant of the square lattice Z^2 (self-avoiding walk growth constant, A387897): rigorous bracket vs 12-digit numerical estimate; #528 headline asks to determine C_k for all k", "kind": "value_gap", "lower": { "value": "2.625622", "year": 2004 }, "upper": { "value": "2.662343", "year": 2022 }, "witness_side": "both", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Numerical value 2.6381585303279... (Jacobsen–Scullard–Guttmann 2016) sits far inside the rigorous bracket — the bracket IS the record. A156816 (positive root of 13x^4-7x^2-581 = 2.63815853034...) is a folklore quartic guess with no known derivation, DOUBTED: disagrees with JSG16 numerics around the 11th decimal (Bauerschmidt et al. 2012 note, in OEIS) — hence exact=wall (no closed form expected). The upper-bound lane looks genuinely movable: Couronné 2022 was a single-author modest-compute improvement (2.679->2.662); longer loops = more compute, same method. General k: trivially k <= C_k <= 2k-1; Kesten 1963: C_k = 2k-1-1/(2k)+O(1/k^2). Honeycomb lattice (not Z^k) solved exactly by Duminil-Copin–Smirnov." }, { "problem": 530, "oeis": "A143824", "quantity": "l(N) — largest m such that EVERY N-element set of reals contains a Sidon subset of size m; asked: is l(N) ~ N^{1/2}?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known: N^{1/2} << l(N) <= (1+o(1))N^{1/2} — lower by Komlós–Sulyok–Szemerédi 1975 (improving the N^{1/3} bound noted by Erdős; problem originally Riddell 1969), upper from the case A={1..N}. That extremal case is exactly A143824 (max Sidon subset of {1..n}), which is why the OEIS relation is marked 'possible' on the site — A143824 is the upper envelope, NOT l(N). Alon–Erdős 1985 stronger conjecture: A decomposes into (1+o(1))sqrt(N) Sidon sets. Page (edited 2026-04-08) cites nothing newer than KSS75 on the main bound; the open content is the constant/order, out of witness reach." }, { "problem": 544, "oeis": "A000791", "quantity": "A000791(10) = R(3,10) — first open cell of the triangle Ramsey row; the machine shadow of #544's asymptotic question R(3,k+1)-R(3,k) -> infinity", "kind": "next_cell", "lower": { "value": "40", "year": null }, "upper": { "value": "41", "year": 2024 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "#544's headline (difference -> infinity; also = o(k)?) is asymptotic and not witness-movable; the R(3,k) table is the finite record feed. R(3,k) ≍ k^2/log k (Kim 1995 + Shearer). erdosproblems page notes a bound from [1014] implying R(3,k+1)-R(3,k) << k^{-c}R(3,k). If no 40-vertex witness exists then R(3,10)=40, and closure needs Angeltveit-class specialized enumeration (not practically DRAT-shaped) — hence exact=unknown, not cell. Witness side has defeated Exoo/McKay/Goedgebeur-grade tooling for ~35 years. Bounds cross-checked OEIS comment (Apr 2024), arXiv v2 (Apr 2024), EJC publication (Nov 2025); no 2026 resolution found in search." }, { "problem": 545, "oeis": "A059442", "quantity": "Erdős–Graham quasi-completeness question: among graphs G with m edges (no isolated vertices), is R(G) always maximized by the quasi-complete graph H (K_n plus a vertex joined to t vertices, m = C(n,2)+t)? No bounded numeric quantity is tracked.", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "The conjecture as literally stated FAILS for small m (2<=m<=5 and 7<=m<=9 — noted by LouisD in erdosproblems comments); open content is the m -> infinity statement. The weaker form R(G) <= 2^{O(sqrt(m))} was proved by Sudakov 2011 (that is #546). A059442 (general Ramsey table) is marked 'possible' related on the site; its own open cells (R(4,6), R(5,5), ...) belong to hub problems 77/78/87, not here — do not double-count them under #545." }, { "problem": 552, "oeis": "A006672", "quantity": "A006672(39) = R(C_4, K_{1,39}) — smallest open cell of the C4-vs-star Ramsey table (the $100 headline asks the asymptotic question: infinitely many n with R(C4,S_n) <= n+sqrt(n)-c?)", "kind": "value_gap", "lower": { "value": "46", "year": 2015 }, "upper": { "value": "47", "year": 1975 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C2", "note": "LESSON BANK (in-repo): our 2026-07-16 'a(39)=46' equality claim was RETRACTED pre-submission — Boza arXiv:2409.12770's '/46' cell is a wheel-convention off-by-one vs DS1 rev.18's open '46-47'; always reconcile against DS1, never a single survey table. OEIS b-file badly lags the literature (ends a(11)=16; exact values known for ALL n <= 38 per Boza's table). Adjacent open cells: a(42) (49 vtx, min-deg 7) and a(44) (51 vtx, min-deg 7); polarity-deletion families ER_7/ER_8 certified UNSAT in-repo for those, so a witness needs a different construction. Our 4 direct SAT deciders on (46,7)/(49,7)/(51,7) were killed after 5.5h with no result (2026-07-17) — naive edge-SAT encodings insufficient; needs symmetry-broken/structured encodings or annealed constructions. Certified in-repo: a(12..16)=17..21 and the a(17)=22 re-derivation (Parsons 1975: R(C4,K1,q^2+1)=q^2+q+2 at q=4; counting vs ex(22;C4)=52 closes the upper side)." }, { "problem": 555, "oeis": "A389313", "quantity": "R_k(C_{2n}) — multicolour Ramsey number of even cycles as a function of the number of colours k (linked OEIS A389313 is the 2-colour diagonal r(C_n,C_n), which is fully solved)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Graham problem: determine R_k(C_{2n}). Known bounds (per problem page, paraphrased): k^{1+1/2n} << R_k(C_{2n}) << k^{1+1/(n-1)} (Erdős 1981); Chung–Graham 1975: R_k(C_4) > k^2-k+1 when k-1 is a prime power, and R_k(C_4) <= k^2+k+1 for all k. The hub's linked OEIS sequence A389313 (Faudree–Schelp 1974, Rosta 1973) is CLOSED — no record lane there. Adjacent machine-shaped record surface lives in Radziszowski DS1's multicolour cycle cells (not verified here). Problem page marks this as not resolvable by finite computation." }, { "problem": 556, "oeis": "A389335", "quantity": "R_3(C_9) — 3-color Ramsey number of the 9-cycle (= A389335(9))", "kind": "bounded_below_only", "lower": { "value": "33", "year": null }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "'The first currently open case' per the cycle-Ramsey survey. No compact CNF for exact-9-cycle-freeness (~2e12 clauses naive) — not even a drat-candidate today." }, { "problem": 573, "oeis": "A006856", "quantity": "A006856(54) — maximum number of edges in a 54-vertex graph of girth >= 5 (first uncomputed cell of ex(n;{C3,C4}))", "kind": "next_cell", "lower": { "value": "185", "year": 2023 }, "upper": { "value": "187", "year": 2026 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Headline problem (Erdős–Simonovits) is asymptotic — is ex(n;{C3,C4}) ~ (n/2)^{3/2}? — and not machine-closable; the linked sequence is the record lane. Exact through n=53: a(50)=175 (unique, Hoffman–Singleton), a(51)=176, a(52)=178, a(53)=181 with >=2688 extremal graphs (McKay). Best known lower bounds a(54..64) >= 185,189,193,197,202,207,212,216,220,224,230 (McKay page, accessed 2026-07-18). Frontier held since Jan 2023 by specialized exhaustive tooling (McKay; SAT approaches by Codish–Miller–Prosser–Stuckey reached order ~32), so a new 186-edge graph is a genuine research result — graded hard. Exact a(54) plausibly falls to the same expert exhaustive methods (cell), but each increment has taken years." }, { "problem": 588, "oeis": "A006065", "quantity": "A006065(19) — maximum number of lines containing exactly 4 points ('4-tree orchard rows') achievable with 19 points in the plane (first cell with a bounds gap in the linked record table)", "kind": "next_cell", "lower": { "value": "20", "year": null }, "upper": { "value": "28", "year": null }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Headline $100 problem is asymptotic — is f_k(n)=o(n^2) for k>=4 (no k+1 points collinear)? — explicitly not resolvable by finite computation. Known (paraphrased from problem page): Kárteszi 1963 f_k(n) >>_k n log n; Grünbaum 1976 f_k(n) >>_k n^{1+1/(k-2)}; Solymosi–Stojaković 2013 (arXiv:1107.0327) f_k(n) >>_k n^{2-O_k(1/sqrt(log n))} — so the answer is delicately between n^{2-o(1)} and o(n^2). The record lane is the linked orchard tables: A006065 exact through n=18 (a(18)=18; Zhao Hui Du verified a(13)-a(16) by code); Friedman gaps: n=19: 20-28, n=20: 23-30, n=21: 24-31, n=22: 26-38. Companion A008997 (exactly-5-point lines): first gap a(15) in [6,9], Berloquin a(21) >= 12. Lower-bound records in this family moved in 2005 (Lin, a(20)>=23) and 2018-2020 (Pegg, n=22 constructions via elliptic-curve/projective tricks) — hence plausible, not open-easy. Exact values are walled by continuous realizability (order-type enumeration stops near n=11). Hub status for this problem is 'wall' (exact side); witness lane is live." }, { "problem": 647, "oeis": "A087280", "quantity": "A087280(6) — the least n > 24 with max_{m infinity. Idén 2026 (Zenodo, CC-BY): search extended 10^10 -> 10^12 with none found; minimum observed gap R(n)-n = 224 near 10^11; empirical gap-growth fit ~0.060(log m)^2.22; independently re-checked a Hughes frontier certificate claimed to cover n <= 6.157x10^17 within the 41 open residue classes (28/30 checks passed — treat the 6.16e17 frontier as claimed, not clean-verified), and confirmed an Alexeev near-witness at n = 604,517,614,941,240. If a term exists, bounded search finds and cheaply certifies it; nonexistence is not machine-provable. Companion sequence A062249 (n + d(n))." }, { "problem": 663, "oeis": "A391668", "quantity": "q(n,k) — least prime not dividing (n+1)(n+2)...(n+k) (OEIS A391668, table read by antidiagonals); conjecture: q(n,k) < (1+o(1)) log n for fixed k >= 2 and n large", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Pomerance problem. Easy bound q(n,k) < (1+o(1)) k log n (also on the OEIS formula lines); the problem asks whether the k factor can be removed, possibly even up to k = o(log n) (a Tao heuristic in the page comments supports this). Companion conjecture: q(n, log n) >= (2+eps) log n infinitely often for some eps > 0. Useful identities from A391668: T(n,k) = A007918(n+k+1) (next prime) for k >= n. Refs: Burr–Erdős–Graham–Li, Acta Arith. 1996; Erdős, Math. Japon. 1997. Problem page: not resolvable by finite computation. See also problem 457." }, { "problem": 668, "oeis": "A385657", "quantity": "A385657(22) — number of nonisomorphic maximally dense unit-distance graphs on 22 vertices (first uncomputed cell of the count sequence linked to this problem)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős 668 asks whether the number of incongruent n-point sets maximizing unit distances tends to infinity, and whether it exceeds 1 for all n > 3 (it equals 1 at n=4). Known counts n=1..21: 1,1,1,1,1,4,1,3,1,1,2,1,1,2,1,1,7,16,3,1,5 (A385657, Weisstein 2025) — caveat from the problem page: computations distinguish graph-isomorphism classes, not congruence classes. Companion A186705 (problem 90's max unit distances) is exact to a(21)=57 with AMP upper bounds for a(22)-a(30) (e.g. best upper for a(27) = 90 per OEIS comment, May 2026). Extending both sequences to n=22 is the same one-step extension of the AMP exhaustive method (arXiv:2412.11914) — machine-shaped but expert-scale, hence exact_feasibility=cell with no cheap-witness lane. Context only: the asymptotic unit-distance conjecture saw major 2026 activity (arXiv:2605.20695, arXiv:2605.20579) — that concerns problem 90, not this counting question. The infinitude question itself is not finite-computation-resolvable." }, { "problem": 669, "oeis": "A003035", "quantity": "A003035(15) — maximum number of lines containing exactly 3 of 15 points in the plane (classical orchard problem; first uncomputed cell)", "kind": "next_cell", "lower": { "value": "31", "year": 2013 }, "upper": { "value": "32", "year": 2013 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Problem 669 as posed asks to estimate f_k(n) (exactly k) and F_k(n) (at least k) and determine lim F_k(n)/n^2, lim f_k(n)/n^2 — asymptotic, not machine-closable; the linked orchard cell is the record lane. Burr–Grünbaum–Sloane 1974: f_3(n) = n^2/6 - O(n); trivially lim F_k(n)/n^2 <= 1/(k(k-1)). Green–Tao 2013 (arXiv:1208.4714): a(n) = floor(n(n-3)/6)+1 for all sufficiently large n, which gives 31 at n=15 — so the 32-line witness likely does NOT exist, and closing the cell probably needs a nonexistence proof over R (abstract 3-configuration enumeration is bounded, but realizability has defeated tools for ~52 years) — hence witness=hard, exact=wall. Also known: a(16)=37, a(17) in {40,41,42}. Discrepancy note: Friedman's maintained table shows ranges 22-24 / 26-27 at n=13,14 where OEIS lists exact 22 / 26; OEIS (Sloane 2013) treated as authoritative here. Cross-links: 4- and 5-point-line versions A006065/A008997 are recorded under problem 588; problem 101 is the k=4 special case." }, { "problem": 676, "oeis": "A390181", "quantity": "Infinitude of A390181 — integers not representable as a*p^2 + b with p prime, a >= 1, 0 <= b < p (equivalently: is every sufficiently large integer so representable? Erdős thought not)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "From the problem page (paraphrased): the sieve of Eratosthenes gives that almost all integers are representable; Brun–Selberg sieve bounds exceptions in [1,x] by << x/(log x)^c. Erdős [1979] called it 'rather unlikely' that all large integers are representable — i.e. he leaned toward the exception set A390181 being infinite (the OEIS comment echoes 'seems likely' infinite). Empirically the exception set is robustly populous: b-file has 100000 terms, the 100000th being 1,211,719 (~8.3% density near 1.2e6). Selfridge–Wagstaff preliminary search suggested infinitude even with the primality condition on p dropped. Sharper variant [Er79d]: minimal c_n with n = a*p^2 + b, 0 <= b < c_n*p; Erdős suggested limsup c_n = infinity and asked if c_n < n^{o(1)}. No gap/record structure to mine." }, { "problem": 681, "oeis": "A389680", "quantity": "Cofiniteness of the property 'there exists k with n+k composite and lpf(n+k) > k^2' — equivalently finiteness of the exceptional set A389680", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős [Er79d], related to Erdős–Eggleton–Selfridge; possibly true with k^2 replaced by k^d for any d. Fresh verification computation (this survey, 2026-07-18; numpy spf-sieve, cross-checked exactly against all 68 OEIS-displayed terms of A389680): counts of exceptional n <= 10^e for e=2..7 are 91, 746, 5712, 40282, 257940, 1441896 — density shrinking by ~x0.63 per decade to 0.1442 at 10^7; largest exception <= 10^7 is 9,999,973. So the OEIS comment 'density decreases' is confirmed but extinction (finiteness) is far beyond any current computational frontier — the problem page correctly marks it not resolvable by finite computation. No gap/record lane exists; a future rung could be 'first exception-free decade' but nothing of the sort is in sight." }, { "problem": 683, "oeis": "A006530", "quantity": "Growth of P(binom(n,k)) (largest prime factor of binomial coefficients) — conjectured lower envelope min(n-k+1, k^{1+c}) for some c>0", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Best known: Sylvester-Schur gives P(binom(n,k)) > k for k <= n/2; Erdos 1955 [Er55d] proved P(binom(n,k)) >> k log k for k <= n/2 (still the record per erdosproblems.com/683, page last edited 2025-12-31). Conjecture asks k^{1+c}; prime-gap heuristics suggest exp(c*sqrt(k)) (essentially equivalent to hub #961). Listed OEIS ids are column/special cases: A074399 = gpf of n(n+1), A121359 = gpf of C(n+2,3), A006530 = generic gpf. Statement from Apache-licensed Lean stub; site prose paraphrased only." }, { "problem": 684, "oeis": "A392019", "quantity": "Growth order of f(n) = least k such that the k-smooth part u of binom(n,k) exceeds n^2 (A392019)", "kind": "value_gap", "lower": { "value": "f(n) >= (1/2 - o(1))·log n for infinitely many n", "year": 2026 }, "upper": { "value": "f(n) = O((log n)^2) for all n", "year": 2026 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "HOT in 2026 — recheck before any use as best-known. Superseded: Tang-ChatGPT f(n) <= n^{30/43+o(1)} (and 2/3 on RH). Newer preprints (both unrefereed, found in fresh search, not yet on the curated page as of its 2026-04-01 edit): Bae arXiv:2604.23784 claims limsup f(n)/log n = infinity (would strengthen the lower side beyond any C log n); Li arXiv:2606.08216 claims density-one asymptotic f(n) = (2/(1-gamma)+o(1))·log n ≈ 4.7305 log n for a.e. n. Mahler's theorem gives f(n)->infinity ineffectively. A392019 (Beregovsky, 2025-12) has 73 terms; a(n)=0 when no k works (small n)." }, { "problem": 687, "oeis": "A048670", "quantity": "A048670(65) = j(p_65#) — Jacobsthal function of the product of the first 65 primes (first uncomputed cell; a(64)=1110)", "kind": "next_cell", "lower": { "value": ">= 1110", "year": 2021 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "$1000 problem. The prize question is the asymptotic gap and is a wall: Y(x) << x^2 (Iwaniec, via linear sieve, Acta Arith. 1971/1978) vs Y(x) >> x·log x·logloglog x/loglog x (Ford-Green-Konyagin-Maynard-Tao, J. AMS 31 (2018)); Maier-Pomerance conjecture x(log x)^{2+o(1)}; fresh search found no post-2018 improvement. Machine lane = the table: exact a(62)-a(64) needed Google-Cloud-scale search (Bozek 2021), so exact a(65) is cell-in-principle but likely months of CPU; the cheap record lane is new covering witnesses lower-bounding a(65) (algorithmics: Ziller-Morack arXiv:1611.03310; Hagedorn Math. Comp. 2009 for h(n), n<50). Identities: A048670(n) = A058989(n)+1 (checked: 1110=1109+1) = 2·A072752(n)+2; A049300 = minimal starting u." }, { "problem": 693, "oeis": "A391118", "quantity": "Asymptotic order of the max gap between consecutive integers in [n, n^k] having a divisor in (n, 2n) (k=2 table: A391118); is it (log n)^{O(1)}?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "No nontrivial proven bounds are cited on erdosproblems.com/693 (only a cross-ref to #446); trivial upper: max gap <= ~n+2 since consecutive multiples of a fixed d in (n,2n) are <= 2n apart. A391118 (Beregovsky, created 2026-01) has terms n=3..83, values reaching only 12 — consistent with polylog but non-probative. Table extension is easy compute, not record-shaped; no bound moves." }, { "problem": 695, "oeis": "A061092", "quantity": "g(k) = minimal terminal prime q_k over ascending prime chains q_1<...= 3·2^(k-2) (each step forces q_{i+1} >= 2·q_i + 1)", "year": null }, "upper": { "value": "g(k) <= A061092(k) (greedy chain 2,3,7,29,59,709,2837,22697,590123,...; known to k=100, a(100) has 175 digits)", "year": 2001 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Greedy is pointwise-minimal per step but need not be globally optimal (a larger early prime can enable much smaller later ones) — bounded DFS over primes ≡ 1 mod q below the greedy cap is an hours-scale search for k <= ~12, and exact g(k) for modest k falls to exhaustive search below the greedy bound. CAVEAT (honesty): Erdos's question concerns one infinite nested chain; g(k)-optimal chains need not nest, so records are evidence, not resolution — lim q_k^{1/k} = infinity? and existence of a chain with q_k <= exp(k(log k)^{1+o(1)}) both remain walls. Greedy grows like exp(exp(O(k))) via Linnik; the conjectured least prime ≡ 1 mod p being <= p(log p)^{O(1)} would give exp(k(log k)^{1+o(1)}). Finite prime chains studied by Ford-Konyagin-Luca (Prime chains and Pratt trees, GAFA 2010). I verified greedy steps a(1..9) by direct computation." }, { "problem": 699, "oeis": null, "quantity": "largest N such that for every n < N and every 1 <= i < j <= n/2 some prime p >= i divides gcd(C(n,i), C(n,j)) -- exhaustive general-row verification", "kind": "verified_range", "lower": { "value": "1e8", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C2", "note": "Believed true, so the deliverable is a verified-to-N record. The sweep rests on a prime-gap pruning lemma (a prime in (n-i,n] settles the row, so only i <= n - prevprime(n) can host a counterexample) and on a THEOREM that large primes never suffice: on every pruned row max S(n,i) <= n/2, so at j = floor(n/2) there is never a covering prime above j -- p=2 can be the unique usable prime. Partial theory: Liam Price (accepted 2026-07-18) covers j <= 3i/2 and n = 2j, about a third of the (i,j) plane. COST CALIBRATION worth keeping: 141.6 us/row solo at 1e6, 897.5 us/row solo at 5e7, 4781 us/row under 13-way parallelism -- estimate at the scale AND concurrency you will actually run at." }, { "problem": 700, "oeis": "A091963", "quantity": "f(n) = min_{1sqrt(n), and f(n) <<_A n/(log n)^A", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdos-Szekeres [ErSz78]. Known per erdosproblems.com/700 (paraphrased): f(n) <= n/P(n) for composite n (take k = exact prime-power divisor), hence f(n) <= (1+o(1))·n/log n; equality f(n)=n/P(n) holds for n=pq (two primes) and n=30; f(n) >= p(n) (smallest prime factor), so n=p^2 gives f(n) >= sqrt(n) — the strict-inequality infinitude for general composite n is the open part. Note OEIS A091963 is phrased as smallest gcd of two interior Pascal-row entries; it coincides with min gcd(n, binom(n,k)) on my tested range 4..43 — flag if extending. Specimen mining (composite non-prime-power n with f(n) > sqrt(n)) could feed the characterization question, exploratory only." }, { "problem": 710, "oeis": "A390246", "quantity": "Growth order of f(n) = least L such that (n, n+L) contains distinct a_1,...,a_n with k | a_k (A390246)", "kind": "value_gap", "lower": { "value": ">= (2/sqrt(e) + o(1))·n·(log n/loglog n)^{1/2}", "year": 1980 }, "upper": { "value": "<= (1.7398...+o(1))·n·(log n)^{1/2}", "year": 1980 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdos offered 2000 rupees (site converts to ~$78) for an asymptotic formula. The gap is the (loglog n)^{1/2} factor plus constants (2/sqrt(e) ≈ 1.213 vs 1.7398...). Freshness: 2026 work by van Doorn (arXiv:2601.16972, Integers 2026) resolves the max_m variant (#711 — intervals of length c·n·log n/loglog n with no system of distinct multiples) but does NOT move #710's 1980 bounds; also arXiv:2603.28636 (van Doorn-Li-Tang 2026) is problem #650, not #710. I verified A390246 = f(n) for n=1..15 by direct matching computation. Cheap non-record analysis: fit the 10^4-term b-file against the two candidate rates to guess the truth — evidence only." }, { "problem": 724, "oeis": "A001438", "quantity": "N(10) — maximum number of mutually orthogonal Latin squares of order 10 (= A001438(10))", "kind": "value_gap", "lower": { "value": "2", "year": 1959 }, "upper": { "value": "6", "year": null }, "witness_side": "lower", "witness_feasibility": "historic", "status": "open", "confidence": "C3", "note": "Open 60+ years whether 3 MOLS(10) exist; 2022–2025 SAT campaigns eliminate relation types only (9 of the Dukes–Howard types remain)." }, { "problem": 725, "oeis": "A001009", "quantity": "A001009 row-12 diagonal = number of reduced 12x12 Latin squares (= A000315(12)) — first uncomputed cell; order 11 = 5363937773277371298119673540771840", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdos's actual #725 question is an asymptotic formula for the number of k x n Latin rectangles: Erdos-Kaplansky 1946 proved ~ e^{-binom(k,2)}·(n!)^k for k = o((log n)^{3/2-eps}); Yamamoto 1951 extended to k <= n^{1/3-o(1)}; the full range is open and not machine-witnessable (I did not exhaustively survey post-1951 range extensions — treat the range frontier as unverified). The machine-shaped frontier is the enumeration table: A000315 has 11 terms (order 11 by McKay-Wanless, 2005), order 12 uncomputed for ~20 years (OEIS keywords hard/more checked live 2026-07-18; fresh web search found no order-12 computation). Not a realistic single-box target — recorded as a wall so the scout deprioritizes." }, { "problem": 727, "oeis": "A343507", "quantity": "A343507(10) = smallest m with (m+10)!^2 | (2m)! — first uncomputed cell (a(9) = 2935782889)", "kind": "next_cell", "lower": { "value": ">= 2935782889", "year": 2021 }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "The Erdos-Graham-Ruzsa-Straus [EGRS75] question (infinitude for each fixed k >= 2) is open even for k=2 and is NOT settled by finite witnesses — the record lane is extending A343507 (keywords hard/more, 9 terms). Growth ratios of a(k) (5x-45x per step) suggest a(10) ~ 10^10-10^11: a days-scale filtered scan on one box. k=1 infinitude is Balakran 1929 (A002503: n with (n+1)^2 | C(2n,n); even perfect numbers m give m-1 as terms). I verified a(2)=208 by direct scan, the equivalence for k=2 with (n+1)^2(n+2)^2 | C(2n,n) (A389396), and the monotonicity implication. Statement from Apache-licensed Lean stub." }, { "problem": 730, "oeis": "A129515", "quantity": "Largest gap k with a known pair (n, n+k) such that binomial(2n,n) and binomial(2(n+k),n+k) have identical prime-divisor sets (sub-question of: are there infinitely many such pairs?)", "kind": "bounded_below_only", "lower": { "value": "2", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "EGRS 1975 said there is 'no doubt' infinitely many pairs exist — that infinitude question is not machine-resolvable. The recordable sub-question (stated open on the problem page): does a pair of shape (n,n+k) exist for every k>=1? Known: k=1 (43 terms listed in A129515, b-file to 10000 terms, all with m=n+1) and k=2 (via the 10003..10005 triple, confirmed here: all three share an identical 1540-prime set). k>=3 is unclaimed. Search cost per n is low but hit-density is unknown (gap-1 pairs are already thin); needs a smart incremental scan, hence 'plausible' not 'open-easy'." }, { "problem": 731, "oeis": "A006197", "quantity": "Asymptotics of m(n) = least integer not dividing binomial(2n,n) (A006197): find f(n) with m(n) ~ f(n) for almost all n", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "EGRS 1975 state it is not hard to show m(n) = exp((log n)^{1/2+o(1)}) for almost all n; the open content is pinning a genuine asymptotic f(n). All values are prime powers (OEIS comment, corrected Eldar 2024). Data-rich but not gap-shaped: no bounded quantity with a cheap witness." }, { "problem": 743, "oeis": "A000055", "quantity": "largest n for which the Gyarfas tree packing conjecture is verified for ALL tuples (T_2..T_n) -- K_n as the edge-disjoint union of trees with |V(T_k)|=k", "kind": "verified_range", "lower": { "value": "10", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C2", "note": "Edge count is exact: sum_(k=2..n)(k-1) = C(n,2), a perfect decomposition with no slack. The hardest tuples at n=10 are those where T_4..T_7 are ALL STARS -- precisely where theory stops, since Gyarfas-Lehel cover the case where all but at most two trees are stars. n=11 multiplies the tuple space by A000055(11)=235. NOTE: erdosproblems.com's [Fi83] key resolves to a DIFFERENT 1983 Fishburn paper (a matrix packing theorem)." }, { "problem": 768, "oeis": "A352287", "quantity": "Leading constant c in the density asymptotic |A ∩ [1,N]|/N = exp(-(c+o(1)) sqrt(log N) loglog N), where A = integers n such that every prime p|n has a divisor d|n, d>1, d ≡ 1 (mod p) (A352287; upper-bounds counts of non-cyclic simple group orders, A001034)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "FRESHNESS FLAG: Eric Li, 'The Sylow Divisor Condition: a Resolution of Erdős Problem 768', arXiv:2606.24872 (2026, 20pp), claims a full resolution with c = 1/(2 sqrt(log 2)); the paper is linked from OEIS A352287 and a claimed solution is posted in the problem-page comments, but upstream still marks the problem OPEN — librarian should track and close if it holds up. Erdős's own bounds: density >= exp(-c sqrt(log N) loglog N) and <= exp(-(1+o(1)) sqrt(log N loglog N)) (paraphrased from problem page). Companion sequence A001034 (simple group orders) is the motivation." }, { "problem": 769, "oeis": "A014544", "quantity": "c(3) — least k0 such that the cube can be decomposed into k homothetic cubes for every k >= k0 (Hadwiger cube-dissection problem; achievable k listed in A014544)", "kind": "value_gap", "lower": { "value": "15", "year": 2018 }, "upper": { "value": "48", "year": null }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Literature is murky: Scott (AMM 54, 1947) reportedly showed the largest non-achievable k is 47 or 54, and Rychener & Zbinden (1977) dissected 54, which would give c(3)=48 exactly — but Hickerson's 2007 OEIS notes could not confirm proofs for 31,32,42,44,47 (impossibility of those five settles the whole list), so the community (and erdosproblems.com, citing Meier's conjecture c(3)=48) still treats it as open. LIBRARIAN TARGET: pull Scott 1947 + Gardner 1992 to possibly close c(3)=48. The Erdős problem is really about general n: is c(n) >> n^n? Known c(n) <= e^2 n^n, c(n) <= 1.8 n^(n+1) for n+1 prime (Connor–Marmorino 2018), c(n) << (2n)^(n-1) (Hudelson 1998), c(2)=6 (all paraphrased from problem page)." }, { "problem": 770, "oeis": "A263647", "quantity": "Distributional questions for h(n) = extremal b such that 2^n-1, 3^n-1, ..., b^n-1 are mutually coprime: do densities delta_p = dens{n : h(n)=p} exist; is liminf h(n) infinite; does the largest prime p with p-1 | n, p > n^eps determine h(n)?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known easy facts (paraphrased from problem page): h(n) = n+1 iff n+1 prime; h(n) unbounded on odd n; plausibly h(n)=3 infinitely often. The attached sequence A263647 (n with gcd(2^n-1, 3^n-1)=1, b-file to 10000 by R. Israel) is actually the sequence OEIS ties to the sibling problem #820 (its infinitude is the Ailon–Rudnick 2004 question); the hub attaches it here as the h(n) >= 4 indicator. No record lane." }, { "problem": 773, "oeis": "A390813", "quantity": "A390813(69) — size of the largest Sidon subset of the first 69 positive perfect squares (first uncomputed cell; table verified to n=68)", "kind": "next_cell", "lower": { "value": "32", "year": 2025 }, "upper": { "value": "33", "year": 2025 }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The Erdős/Alon–Erdős 1985 question — is max Sidon subset of {1,4,...,N^2} of size N^(1-o(1))? — is an asymptotic wall: N^(2/3) lower (Lefmann–Thiele 1995) vs N^(1-c/loglog N) upper (Croot–Mao–Yip, posted in problem-page comments; page edited Apr 2026). The machine lane is the exact table: a(69) falls to bounded search (ILP / max-clique on the Sidon-conflict graph), and each new row is a small verified record. Sievers note: up to n=68 only n=32 fails to attain the max with a subset containing n^2." }, { "problem": 779, "oeis": "A005235", "quantity": "Verified range of Deaconescu's conjecture: for every n>1 with P = primorial(n), some prime p with p_n < p < P has P + p prime (falsifiable; checked for all n <= lower.value)", "kind": "verified_range", "lower": { "value": "1000", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Cheap record available: Fortunate numbers a(n) = A005235(n) satisfy gcd(a(n), P)=1, so a(n) < p_(n+1)^2 forces a(n) prime (Ordowski, OEIS comment) — and then a(n) itself is a Deaconescu witness for n. The b-file reaches n=4600 (Batalov; first 2000 terms verified prime, Arndt 2013), so auditing the b-file with the p_(n+1)^2 test (plus direct primality for any stragglers) should push the verified range 1000 -> ~4600 at near-zero compute. Proving the conjecture for all n is far beyond reach (needs Cramér-strength input); Erdős expected the least such p to be n^O(1) (paraphrased)." }, { "problem": 786, "oeis": "A143301", "quantity": "Maximal density of A ⊆ {1..N} (resp. A ⊆ N) in which products of r elements never equal products of s elements with r ≠ s", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "State (paraphrased from problem page): density version answered negatively — any valid A has density <= 1/2, and upper density <= 1-c with the same c; constructions: Selfridge 1/e - eps, Tao's one-large-prime-factor set ~0.8285 N for the interval version. What stays open is the variant where the products must use distinct elements (Er80 formulation). The attached OEIS entry is just the decimal expansion of the Hall–Montgomery constant (105 digits) — no table to extend, no witness lane." }, { "problem": 791, "oeis": "A066063", "quantity": "A066063(51) — minimal size of S ⊆ {0..51} with S+S ⊇ {0..51} (finite additive 2-basis; first uncomputed cell, table + b-file end at n=50)", "kind": "next_cell", "lower": { "value": "12", "year": 2001 }, "upper": { "value": "13", "year": 2001 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The Erdős/Rohrbach question itself is asymptotic: g(n) ~ 2 sqrt(n) was disproved (Mrose 1979, g(n)^2 <= 3.5n); current frontier g(n)^2/n in [2.181 (Yu 2015), 294/85 = 3.4588 (Kohonen 2017, arXiv:1606.04770-line of work)] — all paraphrased from problem page and cross-checked against arXiv search (no post-2017 h=2 improvement found; arXiv:2507.23627 (2025) improves large-h postage-stamp, not h=2). Beating Kohonen's construction constant would be a genuine record but is expert-search-defeated (grade hard); the honest hours-scale lane is table extension a(51), a(52), ... (keyword 'more' in OEIS)." }, { "problem": 812, "oeis": "A059442", "quantity": "Growth of consecutive diagonal Ramsey gaps: is R(n+1)/R(n) >= 1+c for some c>0, and is R(n+1)-R(n) >> n^2?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Best known toward the difference question (paraphrased from problem page): Burr–Erdős–Faudree–Schelp 1989 give R(n+1)-R(n) >= 4n-8 for n >= 2, and the lower-bound machinery behind #165 gives R(n+2)-R(n) >> n^(2-o(1)) — a two-step, not one-step, gap. Even R(n+1)-R(n) >= cn^2 for a single step is open, and no witness object exists at any finite n. Cross-check: only 9 nontrivial exact diagonal/table values known in A059442 (through R(4,5)=25 etc.); the array is 'hard' keyword." }, { "problem": 820, "oeis": "A263647", "quantity": "A263647 extension — integers n with gcd(2^n-1, 3^n-1) = 1 (equivalently H(n)=3 in Erdős' notation); b-file ends at a(10000) = 24898", "kind": "next_cell", "lower": { "value": "24898", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The actual open problem — is H(n)=3 infinitely often, and does H(n) = exp(n^{(c+o(1))/loglog n}) — cannot be settled by finite computation (wall). Known: Erdős (1974) proved H(n) > exp(n^{c/(loglog n)^2}) for infinitely many n; a sharpening to exp(n^{c/loglog n}) is sketched in an unrefereed site comment (van Doorn). H(n) for n=1..10 is 3,3,3,6,3,18,3,6,3,12 (numeric facts paraphrased from erdosproblems.com/820). A263647 conjectured infinite (Ailon–Rudnick, per OEIS comment). Machine lane = table extension only; it is a bookkeeping rung, not a bound-mover. Grade honest: extending the b-file is hours of compute; the conjecture itself is a wall." }, { "problem": 821, "oeis": "A014197", "quantity": "limsup_n log g(n)/log n, where g(n) = #{m : phi(m) = n} (A014197) — the totient-multiplicity exponent", "kind": "value_gap", "lower": { "value": "0.71568 (= 1 - 15/(32*sqrt(e)))", "year": 2022 }, "upper": { "value": "1", "year": null }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős conjectured the exponent is 1 (g(n) > n^{1-eps} infinitely often). History: Pillai limsup g = infinity; Erdős 1935 g(n) > n^c i.o.; Baker–Harman 1998 was the previous best exponent; Lichtman 2022 smoothness threshold 0.2843–0.2844 gives 0.71568. Freshness: erdosproblems.com/821 (2025-10) cites Lichtman as current best; no 2026 improvement found in searches. Progress requires prime-counting theorems about smooth p-1, not witnesses — pure wall for the records lane." }, { "problem": 830, "oeis": "A259180", "quantity": "infinitude of amicable pairs, and whether the count A(x) of pairs below x satisfies A(x) > x^{1-o(1)}", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Pure existence/density conjecture — not gap-shaped. State of computation: ALL amicable pairs with smaller member < 10^21 are known — the Chernykh 'Amicable Numbers' BOINC exhaustive search finished 2026-04-08 (4,810,863 pairs with 21 digits; sech.me/ap/news.html); database passed 10^9 known pairs back in 2016. State of theory: infinitude is open in both directions; Erdős 1955 A(x)=o(x); Pomerance 2015 A(x) <= x·exp(-(1/2+o(1))(log x · loglog x)^{1/2}) (facts paraphrased from erdosproblems.com/830; Guy B4). New-pair hunting is routine for the BOINC community but mathematically inert here — recommend the scout NOT spend cycles." }, { "problem": 849, "oeis": "A003015", "quantity": "smallest integer a admitting >= 5 solutions of C(n,k) = a with 1 <= k <= n/2 (Singmaster / Erdős–Gordon multiplicity t=5; the t<=4 cells are occupied by a=2,6,120,3003)", "kind": "bounded_below_only", "lower": { "value": "> 10^60", "year": 2017 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Famous 50+-year target (Singmaster 1971; Erdős credits himself and Gordon), but graded hard-not-historic deliberately: Erdős and Singmaster both believed the answer is NO (multiplicity bounded), so a t>=5 witness is conjectured nonexistent — do not burn GPU on blind search. Known multiplicity ladder (half-triangle convention): t=3 at a=120, t=4 at a=3003; infinitely many a with t=3 via C(n,k)=C(n-1,k+1) (Fibonacci parametric family, A090162), including 61218182743304701891431482520. Best theory: Matomäki–Radziwiłł–Shao–Tao–Teräväinen, arXiv:2106.03335 — at most two solutions once k >= exp((log n)^{2/3+eps}). The Erdős form ('for every t some a has EXACTLY t solutions') has open cells at every t >= 5." }, { "problem": 852, "oeis": "A078515", "quantity": "A079007(31) (records chain of A053597/A078515/A079889) — smallest prime starting a run of 31 pairwise-distinct consecutive prime gaps; current record run length is 30, at p = 196948778371", "kind": "next_cell", "lower": { "value": ">= 196948778371", "year": 2012 }, "upper": null, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Best machine cell in this chunk: the record frontier (~1.97e11) is 2012-era; primesieve sweeps 10^12 in CPU-hours, and a rolling 32-gap distinctness window is trivial code — hours-to-days on one box should extend A079007 by several cells. Orientation: this moves the record table for h(x), NOT the open asymptotics — Erdős asked whether (log x)^c < h(x) and h(x) = o(log x); only h(x) -> infinity is proven (Brun sieve), so the asymptotic problem itself is a wall. Site marks 'claims of partial results in comments' (unverified, not relied on)." }, { "problem": 853, "oeis": "A390769", "quantity": "A390769 extension — r(n) = least even positive integer absent from the first n prime gaps (Erdős' r(x)); b-file holds only 72 terms", "kind": "next_cell", "lower": { "value": "n = 72 (a(72) = 16)", "year": 2025 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Very fresh, very shallow sequence (72 terms) — trivially extendable to n ~ 10^10 with a sieve in hours; the first-occurrence-gap literature effectively determines r up to 2^64 already, so the b-file push is mostly librarian work plus independent verification. The mathematical content is a wall: r(x) -> infinity is equivalent to every even number eventually occurring as a consecutive-prime gap (open, Polignac-adjacent), and r(x)/log x -> infinity is stronger still. Erdős' original statement omitted 'even' — the site notes the correction (paraphrased from erdosproblems.com/853, Er85c)." }, { "problem": 854, "oeis": "A389839", "quantity": "A389839(13) — smallest even integer that is not a difference of consecutive integers coprime to 41# = 304250263527210 (primorial of the first 13 primes); table known only for prime(2)#..prime(12)#", "kind": "next_cell", "lower": { "value": "a(12) = 68 is the last computed cell (for 37#)", "year": 2025 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Directly quantifies Erdős' #854: he first guessed every even t up to the maximal gap occurs, then reported doubts after Lacampagne–Selfridge computations showed failure at 13# = 30030 (paraphrased from erdosproblems.com/854, Ob1) — visible in the data as a(6)=20 missing while the Jacobsthal max gap A048670(6)=22 occurs. The max-gap side is far ahead of the spectrum side: A048670 known through k=64 (a(64)=1110; Bozek 2021, Google-Cloud-scale). a(13) is a genuine frontier cell: period 3.04·10^14, coprime density ~0.145 — days-scale on one strong box with a bitmap wheel + symmetry, GPU-friendly. Second Erdős question (are there >> maxgap many occurring even values?) remains open regardless." }, { "problem": 855, "oeis": "A023193", "quantity": "y* = smallest interval width y with an admissible tuple of more than pi(y) elements (Hensley–Richards violation width; rho*(y) > pi(y) makes the second Hardy–Littlewood conjecture contradict the prime k-tuples conjecture)", "kind": "value_gap", "lower": { "value": "> 2330", "year": 2026 }, "upper": { "value": "3159", "year": null }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The famous conjecture itself (pi(x+y) <= pi(x)+pi(y), 'second Hardy–Littlewood') is expected FALSE but only conditionally refuted: Hensley–Richards 1973/74 + tuples conjecture; Clark–Jarvis 2001 kill the Straus variant similarly; best unconditional is Montgomery–Vaughan 1973 pi(x+y) <= pi(x) + 2y/log y (facts paraphrased from erdosproblems.com/855, edited 2026-04-08 — fresh). No witness settles the conjecture itself (that side is a wall); the machine-movable object is the [2331, 3159] window for y*: Engelsma's site reports exact rho* somewhat beyond the OEIS b-file (unverified here, hence the conservative 2330), and decades of dense-tuple search (incl. polymath8-era tooling) never found a violation below 3159 — grade hard. Exact closure would need Engelsma-style exhaustive rho* to ~3159: conceivable with modern branch-and-bound/SAT, hence 'unknown' not 'wall'." }, { "problem": 860, "oeis": "A058989", "quantity": "h(n) — smallest L such that EVERY interval (m, m+L] contains distinct integers a_1..a_pi(n) with p_i | a_i (system-of-distinct-representatives strengthening of the Jacobsthal function; Erdős–Pomerance)", "kind": "value_gap", "lower": { "value": "(3-o(1))·n, and h(n)/n -> infinity", "year": null }, "upper": { "value": "O(n^{3/2}/(log n)^{1/2})", "year": 1980 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "The bounded quantity is an asymptotic order gap (linear-times-unbounded vs n^{3/2}), so the records lane has nothing to move — wall. The linked OEIS anchors are the NON-distinct analogues (Jacobsthal of primorials A048670, known to k=64; run-length version A058989 to n=64): they lower-bound nothing here directly since h demands distinct representatives. Librarian spawn opportunity: a small exact table of h(p_k) appears to be absent from OEIS entirely — computable for k up to ~9 by scanning m over the primorial period with a matching test (open-easy as a computation, but it creates a new sequence rather than moving a known cell). Facts paraphrased from erdosproblems.com/860; page edited 2025-09-30." }, { "problem": 864, "oeis": "A389182", "quantity": "A389182(101) — max size of A ⊆ {1..101} such that at most one integer has ≥2 representations as a+b (a≤b∈A); first uncomputed table cell (table complete through n=100)", "kind": "next_cell", "lower": { "value": "16", "year": 2025 }, "upper": { "value": "17", "year": null }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Sequence created Oct 2025 — everything past n=100 is fresh territory; branch-and-bound/ILP plausibly extends the table well past 150 in hours. Asymptotic layer (the actual Erdős question): Erdős–Freud [ErFr91] construction gives liminf a(N)/√N ≥ 2/√3 ≈ 1.1547 (Sidon set in [1,N/3] ∪ its reflection); conjectured tight. One-exception sets are quasi-Sidon (only one colliding sum ⇒ |A+A| ~ |A|²/2, containment derived here), so Pikhurko's #840 upper bound (1/4+1/(π+2)²)^{-1/2} ≈ 1.8626 applies [Pi06]; erdosproblems.com lists #864 as a weaker form of #840. Asymptotic gap [1.1547, 1.8626] on the constant is a wall; the table lane is the machine surface." }, { "problem": 878, "oeis": "A339378", "quantity": "f(n) = A339378(n) power-sum (sum over p|n of the largest power of p ≤ n) — asymptotics of max_{n≤x} f(n), of H(x)=Σ_{n H(n) − n^{1+o(1)} where H(n)=Σ_{p 1163 (= a(6); all smaller Paley primes fail)", "year": 2023 }, "upper": { "value": "<= 200704 (Graham–Spencer sufficiency ceiling)", "year": null }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Extrapolated a(7) ≈ 4000–6000; brute exact check is days-to-weeks on one box. Needs a symmetry-reduced SAT/CP decision method before it is movable." }, { "problem": 912, "oeis": "A071626", "quantity": "h(n) = A071626(n) — number of distinct exponents in the prime factorization of n!; existence of c>0 with h(n) ~ c·(n/log n)^{1/2}", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Selfridge proved h(n) ≍ (n/log n)^{1/2} [Er82c]. Tao's Cramér-model heuristic (Aug 2025 forum comment, referenced from OEIS A071626 links) predicts c = √(2π) ≈ 2.5066 (per erdosproblems.com/912, paraphrased). DISCREPANCY FLAG: the hub stub carries formal-conjectures formal_status 'research solved', but erdosproblems.com/912 showed OPEN when fetched 2026-07-18 — treat as open; the Lean file plausibly formalizes the solved ≍ statement rather than the ~ conjecture. B-file to 10^4 (Corneth) enables constant-testing numerics only." }, { "problem": 913, "oeis": "A359747", "quantity": "A359747 verified frontier — largest verified k such that k(k+1) has all-distinct exponents in its prime factorization (empirical support for the infinitude conjecture)", "kind": "verified_range", "lower": { "value": "32327407", "year": 2023 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Every term has k or k+1 powerful (OEIS comment), so terms are Pell-sparse but easy to enumerate. Conditional resolution (per erdosproblems.com/913, paraphrased): infinitely many primes p with 8p²−1 also prime would settle it affirmatively via n=8p²−1 (exponents {1,2,3}) — a Schinzel-type hypothesis, i.e. the theoretical side is a wall. Table extension and the both-powerful side-hunt are the cheap lanes; neither closes the problem." }, { "problem": 928, "oeis": "A006530", "quantity": "natural density of {n : P(n) < n^α and P(n+1) < (n+1)^β} for fixed α,β ∈ (0,1) — existence (conjectured value ρ(1/α)·ρ(1/β), Dickman ρ)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "State of the art (per erdosproblems.com/928, paraphrased; page last edited 2026-04-03): Teräväinen [Te18] proved the LOGARITHMIC density exists and equals ρ(1/α)ρ(1/β); Wang [Wa21] gets the natural density conditionally on Elliott–Halberstam for friable integers; single-condition density is Dickman 1930. Open content: unconditional natural density. Infinitude of simultaneous smooth pairs follows from Schinzel [Sc67b] (Meza observation). No record lane." }, { "problem": 932, "oeis": "A387864", "quantity": "A387864 verified frontier — largest verified r such that at least two integers strictly between p_r and p_{r+1} are (p_{r+1}−p_r)-smooth (empirical support for the infinitude conjecture)", "kind": "verified_range", "lower": { "value": "6386830", "year": 2025 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Erdős [Er76d] believed it true but with density-0 r (he proved density 0 even for one such n); empirically abundant regardless — 10^4 terms below 6.4e6. Tao is among those flagging the problem 'difficult' on the page (fetched 2026-07-18). Lean-formalized (research open). Sequence created Sep 2025 (Radcliffe); frontier extension is a cheap standing lane but cannot close the problem." }, { "problem": 935, "oeis": "A057521", "quantity": "E_2 = limsup_n log_n Q_2(n(n+1)(n+2)) — powerful-part exponent over 3 consecutive integers (ℓ=2 case of #935; Q_2 = A057521 powerful part)", "kind": "value_gap", "lower": { "value": "2", "year": 2026 }, "upper": { "value": "3", "year": null }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Erdős's main question asserts E_2 = 2 (Q_2 < n^{2+ε} eventually for every ε), which he called very difficult; ABC implies the weaker Q_2 = o(n^{ℓ+1}) (question 3) per [Fe26] (all per erdosproblems.com/935, paraphrased; page last edited 2026-02-08). Question 2 (limsup Q_2/n² = ∞ for ℓ≥2) is RESOLVED affirmatively via the x²−8y²=1 Pell family (van Doorn / [Fe26], essentially #367). For ℓ=1 the exponent question is closed trivially: Q_2(n(n+1)) ≤ n(n+1) and consecutive powerful pairs (A060355) attain it, so E_1 = 2 exactly (derived here); A389244 tabulates the ℓ=1 values. Second linked sequence: A389244." }, { "problem": 936, "oeis": "A146968", "quantity": "Brocard sub-case search frontier: largest N such that no n in (7, N] has n!+1 a perfect square (A146968 = {4,5,7}); the erdosproblems statement asks whether 2^n±1 and n!±1 are powerful only finitely often", "kind": "verified_range", "lower": { "value": "1e12", "year": 2019 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Range extension is sequential-per-prime but parallel-across-primes: ~1e12 mulmods per prime chain = hours on one box. Only the square sub-case is machine-frontier; testing n!±1 or 2^n±1 for POWERFUL requires full factorization (no cheap search; 2^n±1 factor state = Cunningham tables). Known powerful value: 2^3+1=9. Conditional results: Cushing–Pascoe 2016 (abc ⇒ finitely many powerful values near n!), CrowdMath 2020 (abc ⇒ same for 2^n±1), per erdosproblems #936 remarks. Wasserman 2005: no square n!+1 values < 10^20000. Matson's 1e12 is a claim reported in OEIS, not a refereed publication — treat as soft frontier; 4e9 (Peters) is the conservative rigorous line." }, { "problem": 938, "oeis": "A001694", "quantity": "3-term arithmetic progressions of CONSECUTIVE powerful numbers n_k, n_{k+1}, n_{k+2} (A001694): complete table up to search limit; largest known example", "kind": "verified_range", "lower": { "value": "1e12 (search limit; exactly 10 triples exist with all terms ≤ 1e12, largest = (697592448400, 697593096200, 697593744000), gap 647800)", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The 10 triples ≤ 1e12: (1728,1764,1800), (6912,7056,7200), (729000,729316,729632), (1458000,1458632,1459264), (2916000,2917264,2918528), (11664000,11669056,11674112), (149022674775,...), (260102040004,...), (348796224200,...), (697592448400,...). NOT in OEIS as of 2026-07-18 (searched 697592448400 and initial terms — no hits): candidate new sequence + submission. Scaling a triple by a square preserves AP+powerful but consecutiveness must be re-verified, so infinite families are not automatic. Erdős's finiteness question itself cannot fall to computation (abc-flavored). Context: Chan, J. Int. Seq. 26 (2023) 23.1.1 (APs among powerful numbers, not-necessarily-consecutive); Chan arXiv:2503.21485 (n,n+1,n+2 variant, cf. Erdős #364). Extending the exhaustive limit to 1e14–1e16 is minutes-to-hours (O(sqrt(X)) powerful numbers)." }, { "problem": 945, "oeis": "A048892", "quantity": "A048892(19) — smallest start of 19 consecutive integers with pairwise distinct numbers of divisors (first uncomputed cell; F(a(n)) ≥ n feeds the Erdős–Mirsky F(x) ≤ (log x)^{O(1)} question)", "kind": "next_cell", "lower": { "value": "239919791836864", "year": 2023 }, "upper": null, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "a(16)=3.47e11, a(17)=2.62e12, a(18)=2.40e14 — extrapolation suggests a(19) ~ 1e16, i.e. Ehrenstein-class dedicated sieving (weeks), hence graded hard despite the trivially cheap verifier. Underlying asymptotics (Erdős–Mirsky 1952): (log x)^{1/2}/loglog x ≪ F(x) ≪ exp(O((log x)^{1/2}/loglog x)); erdosproblems #945 credits Beker with upper F(x) ≪ exp(O((log x)^{1/3+o(1)})) — primary source NOT located on arXiv (2026-07-18 search), treat as site-remark attribution; Cambie: Cramér ⇒ F(x) ≪ (log x)^2. Exact cell falls to bounded exhaustive scan + cheap verify exactly as a(17), a(18) did." }, { "problem": 954, "oeis": "A390642", "quantity": "error term E(x) = #{(i,j): a_i + a_j ≤ x} − x for Rosen's greedy sequence (A390642); Erdős asks if E(x) = O(x^{1/4+o(1)})", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "By construction the solution count is ≥ x; Erdős and Rosen could not even prove it is ≤ (1+o(1))x (per erdosproblems #954 remarks). OEIS A390642 is brand new (created 2025-12-01, 55 terms, no b-file) with the comment-conjecture a(n) ~ (pi/8) n^2. No literature bounds found. Scout value: numeric exponent-fit at scale is a cheap exploratory (non-record) task; extending the OEIS sequence + b-file is a trivial concrete contribution." }, { "problem": 961, "oeis": "A213253", "quantity": "A213253(269) — first cell beyond Najman's computed range n=1..268; f(n) = minimal L such that every L consecutive integers > n contain one with a prime factor > n (equivalently 1 + longest run of n-smooth integers > n)", "kind": "next_cell", "lower": { "value": "16", "year": 2011 }, "upper": { "value": "269", "year": 1892 }, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "b-file (268 terms) unchanged since it was entered from Najman's paper; no extension located (2026-07 search). The EXACT value needs a nonexistence proof over all integers — Najman's bounded Lehmer/Pell (Størmer-type) machinery achieved this for n ≤ 268 but its scaling past there is unclear to me, hence exact_feasibility unknown rather than cell. Asymptotics: f(k) < 3k/log k (Erdős 1955); f(k) ≪ (logloglog k/loglog k)·k/log k (Jutila 1974; Ramachandra–Shorey 1973); conjectured (log k)^{O(1)}, heuristically (log k)^2 (Najman remark in A213253). Site marks this essentially equivalent to erdosproblems #683. Adjacent fresh work: van Doorn–Tang arXiv:2606.19863 (June 2026) treats runs avoiding primes in (k,2k) — related family, not this quantity." }, { "problem": 962, "oeis": "A327909", "quantity": "A327909(1000) — smallest start of a run of 1000 consecutive integers each having a prime factor > 1000 (first cell beyond the 999-term b-file; per-n table underlying Erdős's k(n))", "kind": "next_cell", "lower": { "value": "22369305365", "year": null }, "upper": null, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "a(999) ≈ 2.24e10 and the sequence grows slowly (long constant stretches), so a segmented 1000-smooth sieve to ~1e11 (≈ 2e11 division ops in C) settles a(1000) in hours on one box — among the easiest genuine cells in this chunk. Asymptotics for Erdős's k(n): log k(n) ≫ sqrt(log n loglog n) (Erdős 1976, per erdosproblems #962); constant improved to (1/sqrt(2) − o(1)) attributed to Tang in site remarks (2026) — primary arXiv source NOT located, treat as site attribution; upper: k(n) ≤ (1+o(1))sqrt(n) (Tao, site comment); Erdős reported exp(−(log n)^c)·sqrt(n). Extending the table gives data for the log k(n) = (log n)^{1/2+o(1)} question. b-file year unrecorded (entry rev 2025-10-28), hence year null." }, { "problem": 968, "oeis": "A387591", "quantity": "lower density of {n : p_n/n < p_{n+1}/(n+1)} (the index set behind A387591) — conjectured positive", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Prachar 1961 [ErPr61] proved the COMPLEMENT {u_n > u_{n+1}} has positive (lower) density and that sum over p_n < x of |u_{n+1} − u_n| ≍ (log x)^2 (per erdosproblems #968 remarks). Erdős also asked for infinitely many monotone triples u_n < u_{n+1} < u_{n+2} and the reverse. Tao has flagged the problem 'looks difficult' on the site (2026). A387591 is new (Sep 2025). No bounded record quantity identified." }, { "problem": 969, "oeis": "A013928", "quantity": "exponent theta = inf{t : E(x) ≪ x^t} of the error term E(x) = Q(x) − (6/pi^2)x in squarefree counting (A013928 is Q's counting function)", "kind": "value_gap", "lower": { "value": "1/4", "year": 1931 }, "upper": { "value": "1/2 (elementary; best unconditional is x^{1/2−o(1)}-type savings, Walfisz; on RH: 11/35 + o(1), Liu)", "year": 1963 }, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "E(x) ≪ x^{1/4+o(1)} would IMPLY RH (per erdosproblems #969), and even under RH the true order is unknown — a definitional analytic wall for the records lane. Believed true order: x^{1/4}. PNT already gives o(x^{1/2}). Schnirelmann-density curiosity in A013928: min a(n)/n = 53/88 at n=176 (Rogers 1964)." }, { "problem": 970, "oeis": "A048669", "quantity": "A048670(65) — Jacobsthal function g of the product of the first 65 primes (first uncomputed primorial cell; the lower-bound engine for Erdős's h(k) and the h(k) ≪ k^2 question)", "kind": "next_cell", "lower": { "value": "1110", "year": 2021 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Exact a(62)–a(64) required Google-Cloud-scale search (Bozek 2021); expert tooling is Ziller–Morack arXiv:1611.03310 + Ziller arXiv:1903.11973 — a(65) exact is bounded-exhaustive but heavy; good greedy/covering lower-bound runs are a hours-to-days heuristic-search game. CAVEAT for the Erdős quantity: Hajdu–Saradha (Math. Comp. 81, 2012) disproved Jacobsthal's conjecture that primorials maximize — at k=24 some 24-prime-factor n has g = 236 > a(24) = 234 — so A048670 lower-bounds h(k) but is not h(k) itself. Asymptotics: Iwaniec 1978 h(k) ≪ (k log k)^2; FGKMT (arXiv:1412.5029, JAMS 2018) g(p_k#) ≫ k log k logloglog k/loglog k; Pintz-type refinement with constant 2e^gamma in A048670 comments; Maier–Pomerance heuristic suggests a(n) = n(log n)^{3+o(1)}, i.e. Jacobsthal's k^2 conjecture as stated in #970 is expected FALSE at the (log k) level while h(k) ≪ k^{1+o(1)}... note carefully: heuristic n(log n)^3 is consistent with h(k) ≪ k^2. Stub lists A048669 (g(n) itself, easy, b-file 1e4); the record cell lives in A048670." }, { "problem": 971, "oeis": "A226521", "quantity": "existence of c > 0 with #{a mod d : p(a,d) > (1+c) phi(d) log d} ≫ phi(d) for ALL large d, where p(a,d) = least prime ≡ a (mod d) (A226521 tabulates p(a,d))", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős [Er49c] proved the statement for an infinite sequence of d, and complementarily that for any eps > 0, p(a,d) < eps phi(d) log d holds for ≫_eps phi(d) residues (per erdosproblems #971 remarks). The heuristic mean of p(a,d)/(phi(d) log d) is ~1; the question is a uniform fat-upper-tail claim. No bounded record quantity: numeric d-sweeps (say all d ≤ 1e6 with several c values) would be cheap supporting data only." }, { "problem": 975, "oeis": "A147807", "quantity": "existence of the asymptotic constant c(f) in Sum_{n<=X} tau(f(n)) ~ c*X*log X for irreducible f in Z[x] of degree >= 3", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Order of magnitude is settled: Sum tau(f(n)) >> X log X (van der Corput 1939) and << X log X (Erdős 1952, elementary), so only the existence of the limit constant is open, and only for deg f >= 3. All irreducible quadratics are solved (Hooley 1963; McKee 1995/1997/1999 give the constants), e.g. Sum_{n<=x} tau(n^2+1) = (3/pi) x log x + O(x). A147807 tracks the SOLVED quadratic case (partial sums of tau(n^2+1)/2, c = 3/(2*pi) per Eldar's 2023 OEIS comment) — it is a sanity harness, not a frontier. No bounded quantity with a movable record exists here." }, { "problem": 979, "oeis": "A385316", "quantity": "A385316(6) — smallest N that is a sum of 3 prime cubes in exactly 6 ways", "kind": "next_cell", "lower": { "value": "> 10^12", "year": 2026 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "record_in_progress", "confidence": "C2", "note": "PUBLIC BOUND: a(6) > 10^12, replayable from certificates/erdos-979/ (~11 GB). PRIVATE OPERATIONAL RESULT: the private streaming sweep and blind replica report gap-free agreement through 10^13, but their code and ledgers are not tracked in this repository, so that stronger result is quarantined from public promotion pending a self-contained replay packet. External OEIS submission is a separate human-sent step." }, { "problem": 982, "oeis": "A004526", "quantity": "f(n)/n where f(n) = largest m such that every convex n-gon has a vertex with >= m distinct distances to the other vertices (Erdős 1946 conjecture: f(n) = floor(n/2))", "kind": "value_gap", "lower": { "value": "(13/36 + 1/22701)*n - O(1)", "year": 2013 }, "upper": { "value": "floor(n/2)", "year": 1946 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Lower-bound chain (proof-only, not witnessable): ceil(n/3) Moser 1952 -> floor(n/3+1) Erdős–Fishburn 1994 -> ceil((13n-6)/36) Dumitrescu 2006 -> (13/36 + 1/22701)n - O(1) NPPZ 2013, still current per Sheffer's distinct-distances survey (arXiv:1406.1949) and the problem page (last ed. Oct 2025). The stronger 1946 variant (a vertex with no 3 equidistant co-vertices) is FALSE, so naive attack routes are closed. Witness lane is disproof-shaped: conjecture is widely believed true, so grade hard; but the verifier is trivial, making any claimed counterexample cheap to audit. A004526 = floor(n/2) is just the target formula, not data." }, { "problem": 985, "oeis": "A002233", "quantity": "largest N such that every prime 3 <= p <= N has a prime primitive root q < p (verified range of the conjecture)", "kind": "verified_range", "lower": { "value": "104729", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Edge convention: for p=2 no prime q<2 exists; read for p >= 3 (Guy UPINT F9 phrases it for large p; the stub's Lean form uses q <= p). Literature frontier is well beyond my certified floor: Paszkiewicz–Schinzel (Math. Comp. 71:1307–1321, 2002) tabulated least prime primitive roots and Paszkiewicz (Math. Comp. 78:1193–1195, 2009) computed least primitive roots for ALL p < 10^12 (per Moree's Artin survey arXiv:math/0412262, sect. on g(p)=h(p)); I could not pin the exact certified range of the PRIME-primitive-root tables, so lower is the conservative self-verified 104729 — a librarian pass on P–S 2002 should lift it toward ~10^12. A meaningful new record means N > 10^12 (~3.6e10 primes to process): days-scale but standard, hence plausible not open-easy. Theory side: Heath-Brown 1986 — at least one of {2,3,5} is a primitive root for infinitely many p; full statement is Artin-conjecture-hard, no finite resolution." }, { "problem": 993, "oeis": "A000055", "quantity": "largest n for which every free tree on at most n vertices is verified to have a unimodal independence sequence", "kind": "verified_range", "lower": { "value": "30", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C2", "note": "n=30 alone (14,830,871,802 trees) is 1.71x the ENTIRE cumulative n<=29 workload -- not an incremental rung. Do NOT test log-concavity as a proxy: it is FALSE for trees from n=26 (Kadrawi-Levit-Yosef-Mizrachi arXiv:2305.01784 found exactly two order-26 trees unimodal but not log-concave; arXiv:2510.18826 found tens of thousands more to 101 vertices). The frontier holder is actively working this lane (repo pushed 2026-07-22, cloud compute), so a verified-to-N record here has a short shelf life; the replication half does not." }, { "problem": 1003, "oeis": "A001274", "quantity": "infinitude of {n : phi(n) = phi(n+1)} (the set tabulated by A001274)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Search frontier is FRESH: 42999 solutions below 10^15, exhaustive (Jud McCranie, OEIS comment dated Jan 27 2026); b-file has 42499 terms. Only count-type theory known: #{n<=x} <= x/exp((log x)^{1/3}) (Erdős–Pomerance–Sárközy 1987); reciprocal sum in (1.4325, 7.8358) (Kinlaw–Kobayashi–Pomerance, Acta Arith. 196:69–92, 2020); heuristic a(n) ~ C n^3 log n (Ordowski, OEIS). Adjacent record-shaped cheap-witness targets for the scout (verified by a few factorizations each): (i) a SECOND consecutive pair — only (5186,5187) gives phi(n)=phi(n+1)=phi(n+2) below 10^15 (Lal–Gillard/Firoozbakht question); (ii) a third counterexample to the 'all composite terms are 2 or 3 mod 6' pattern (known exceptions 3044760173455 and 127872660090175). Neither resolves #1003 but both are publishable OEIS-level records." }, { "problem": 1005, "oeis": "A386893", "quantity": "A386893(101) — first OEIS-uncomputed value of f(n), the Mayer–Erdős run-length: largest m such that any two Farey fractions of order n at index distance <= m are similarly ordered (table currently ends at n=100)", "kind": "next_cell", "lower": { "value": "2", "year": 2025 }, "upper": { "value": "30", "year": 2025 }, "witness_side": "both", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "MINER RESULT: a(101) = 27, computed and double-checked here, matching van Doorn's conjectured exact formula f(n) = floor(n/4) + d, d = (1,2,2,4) for n = (0,1,2,3) mod 4, n >= 92. The real lane is bulk: extend the table to n ~ 10^4-10^5 hunting a counterexample to the exact formula (any mismatch is publishable and would kill the conjectured c = 1/4 form), plus a free OEIS b-file extension beyond n=100. Headline asymptotic gap (proof-only): (1/12 - o(1))n <= f(n) <= n/4 + O(1) (van Doorn 2025; history: Mayer 1942 f -> infinity, Erdős 1943 f >> n). CAUTION: the problem page (accessed 2026-07-18) flags a claimed solution posted in comments, not yet incorporated — librarian should re-check status before major effort." }, { "problem": 1013, "oeis": "A292528", "quantity": "A292528(6) — minimum vertices of a triangle-free graph with chromatic number 6", "kind": "value_gap", "lower": { "value": "32", "year": 2017 }, "upper": { "value": "40", "year": 2017 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "[32,40] unmoved since 2017 against Goedgebeur's own world-class enumeration tooling (a(5)=22, Jensen–Royle 1995). Witness side is legitimate but expert-defeated." }, { "problem": 1016, "oeis": "A105206", "quantity": "m(38) — minimum number of edges of a pancyclic graph on 38 vertices; first open cell after Griffin's exact table (hub quantity h(n) = m(n) - n)", "kind": "next_cell", "lower": { "value": "43", "year": 2013 }, "upper": { "value": "44", "year": 2013 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "A genuine two-value cell: m(38) is 43 or 44, decided either way by a bounded search (Griffin ran the analogous 5-chord exhaustion to 31 vertices on 2013 hardware; his Figure-1 construction with 5 chords tops out at exactly n=37, which is why the table stops there). Griffin 2013 gives m(n) exactly for all n <= 37: m(n) = n+5 for 25 <= n <= 37, agreeing with George–Marr–Wallis. OEIS A105206 is STALE — it ends at a(22) = 26 while exact values to n=37 have been published since 2013; the b-file extension n=23..37 (values 27,28,30,31,...,42) is a free micro-contribution. The hub's asymptotic question (is h(n) >= log_2 n + log_* n - O(1)? Bondy 1971 window log_2(n-1) - 1 <= h(n) <= log_2 n + log_* n + O(1)) is a wall, but every table cell is machine-decidable and each new cell probes which side of the window m(n) tracks. Sanity: h(38) in {5,6} vs log_2(37) - 1 ~ 4.2." }, { "problem": 1029, "oeis": "A059442", "quantity": "divergence of R(k)/(k*2^(k/2)) as k -> infinity (Erdős: $100 for proof, $1000 for disproof)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Hub status: wall — correctly so. Known window: R(k) >= (1+o(1))(sqrt(2)/e) k 2^(k/2) (Spencer 1975, still the best lower-order constant; Erdős–Szekeres 1935 gave k*2^(k/2) << R(k) <= binom(2k-1, k-1)) vs upper R(k) <= 3.7992^(k+o(k)) (Gupta–Ndiaye–Norin–Wei arXiv:2407.19026, 2024, optimizing Campos–Griffiths–Morris–Sahasrabudhe arXiv:2303.09521's (4-eps)^k breakthrough). Any lower bound of the form k*2^(k/2)*omega(k) with omega -> infinity resolves the problem — 50 years without improving even the constant. The (sqrt 2)^k-vs-3.8^k chasm is the wall; nothing witness-shaped lives here." }, { "problem": 1030, "oeis": "A000791", "quantity": "liminf R(k+1,k)/R(k,k) — Erdős–Sós: is it > 1 + c for some absolute c > 0?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Known: R(k+1,k) - R(k,k) >= k - 2 trivially; best is the ADDITIVE bound R(k+1,k) - R(k,k) >= 2k - 5 (Burr–Erdős–Faudree–Schelp 1989) — hopelessly far from the multiplicative 1 + c; per the problem page (last ed. 2026-03-23) even R(k+1,k) - R(k,k) > k^c for some c > 1 is unknown. Small-k data (from A059442/A000791, consistent but probative of nothing): R(4,3)=9 vs R(3,3)=6; R(5,4)=25 vs R(4,4)=18; R(5,5) in [43,46] vs R(5,4)=25. Both known asymptotic tools (Spencer-type lower bounds, CGMS/GNNW upper bounds) are too coarse to separate R(k+1,k) from R(k,k) multiplicatively." }, { "problem": 1052, "oeis": "A002827", "quantity": "Number of known unitary perfect numbers (terms of A002827: k with usigma(k) = 2k)", "kind": "bounded_below_only", "lower": { "value": "5", "year": 1975 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "$10 prize (Carlitz/Erdos/Subbarao per Guy B3); Subbarao offered 10 cents per new example. No odd unitary perfect numbers exist (proven; all UPNs even). Constraints on a sixth: Frei 1978 - if not divisible by 3, then divisible by 2^m with m >= 144, >= 144 distinct odd prime factors, and > 10^440. Goto 2007: a UPN with omega(m)=k satisfies m < 2^(2^k). Fifth term found 1975; 50 years without a sixth. Freshness: web search 2026-07-18 found no new discovery. Upstream marks the finiteness question as not resolvable by finite computation." }, { "problem": 1053, "oeis": "A007539", "quantity": "Largest k for which a k-perfect number (sigma(n) = kn) is known", "kind": "bounded_below_only", "lower": { "value": "11", "year": 2023 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The actual Erdos question (Guy B2): for k-perfect n, must k = o(log log n)? Asymptotic wall - but the record ladder is the largest attained abundancy, stuck at k=11 since ~2001. RH-conditional bound (A007539 comment, Greathouse): least n-fold perfect > exp(exp(n/e^gamma)), so a 12-perfect has >= ~10^366. Guy also reports the suggestion that only finitely many k-perfect numbers with k >= 3 may exist. Related table A007539 (least n-fold perfect) has b-file through n=8 (a(8) ~ 8.27e132); minimality of the largest entries rests on census completeness, which Flammenkamp calls extremely probable only for abundancy <= 7 - treat a(8) as least-known. Freshness: search 2026-07-18 found no 12-perfect discovery; PrimeGrid-adjacent efforts ongoing." }, { "problem": 1054, "oeis": "A167485", "quantity": "Asymptotics of f(n) = A167485(n) = least m such that n equals the sum of the k smallest divisors of m: is f(n) = o(n) (everywhere / almost everywhere)? Is limsup f(n)/n infinite?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Guy B2. Proven: f undefined exactly at n=2 and n=5 among small n (Lean textbook theorems in formal-conjectures 1054.lean); OEIS comment conjectures 2 and 5 are the only zeros, which would follow from a strengthened Goldbach statement (every even N > 22 is p+q with 5 < p < q < 5p). Robert Israel: liminf f(n)/n = 0 via f(sigma(n)) <= n. Computed range: b-file to n=20000 (Israel), f(n)/n plotted to 10^6 (OEIS link) - extending these is easy but moves no bound." }, { "problem": 1055, "oeis": "A005113", "quantity": "A005113(20) - least prime in class 20 of the Erdos-Selfridge classification (class via largest-class prime factor of p+1)", "kind": "next_cell", "lower": { "value": "131988573220065", "year": 2007 }, "upper": { "value": "764276710625653", "year": 2007 }, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Underlying open problem (Guy A18, Erdos 1977): are there infinitely many primes in each class, and does p_r^(1/r) tend to infinity (Erdos) or stay bounded (Selfridge)? Known: number of class-r primes <= n is n^(o(1)). Structure for search: any class-20 prime p < 7.6e14 has p+1 = t*q with q a class-19 prime (q >= a(19) = 6.6e13) and even t <= 11, so the bottleneck is enumerating class-19 primes below 3.8e14 - recursive, days not hours. Data ends at a(19) (19 terms, keyword more, no b-file); both the a(20) cell and tightening [1.32e14, 7.64e14] are records. Freshness caveat: bounds are 2007 OEIS comments; no newer source located." }, { "problem": 1056, "oeis": "A060427", "quantity": "A060427(15) - least prime p admitting 15 consecutive intervals of integers each with product congruent to 1 mod p (also: existence frontier k=15 of the for-all-k conjecture)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Guy A15. Existence known: k=2 (Erdos, 1979 letter: intervals 3-4, 5-7 mod 11), k=3 (Makowski 1983 mod 17) per erdosproblems 1056 remarks - but OEIS A060427 already extends verified existence to k=14 (a(14) = 10428007), far beyond what the problem page records. Search = prefix-factorial residue multiplicity: a(n) is the least prime with some residue hit >= n+1 times by j! mod p. a(13)=2374649, a(14)=10428007 (ratio ~4.4) suggest a(15) within 10^8-10^9; O(p) per prime makes the exhaustive scan ~10^15-10^16 ops - GPU-days, exact value falls to the same bounded scan. Empirically a(n) is nondecreasing but no lower bound for a(15) is proven, so lower is left null. Noll-Simmons generalization: arbitrarily many equal factorials mod p. erdosproblems page has 9 comments including claimed partial results - check before attacking." }, { "problem": 1057, "oeis": "A006931", "quantity": "A006931(40) - least Carmichael number with exactly 40 prime factors", "kind": "next_cell", "lower": { "value": "14909796388965607134586226733905214934462755608741300153203070717079045", "year": null }, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "The linked Erdos problem 1057 is the counting asymptotic C(x) = x^(1-o(1)) for Carmichael numbers - an analytic wall, not witness-movable. Bounds per erdosproblems 1057 remarks (page edited 2026-01-16): C(x) > x^0.3389 (Lichtman 2022), improving x^0.33336704 (Harman 2008) and x^(2/7) (Alford-Granville-Pomerance 1994); C(x) < x*exp(-c log x * logloglog x / loglog x) (Erdos 1956), which Pomerance 1989 heuristically matches. The OEIS cell a(40) is the adjacent computable record only. Exact values a(36)-a(39) were computed by Shallue-Webster (arXiv:2506.09903, 2025-26; a(39) ~ 2.475e81 in b-file, found by Butler undergrads Johnson & Webster, added to OEIS Jan 2026) - the lane is ACTIVE; read that paper before spending compute, a(40) may already be in progress or listed there." }, { "problem": 1059, "oeis": "A064152", "quantity": "Infinitude of Erdos primes: primes p with p - k! composite for every k with 1 <= k! < p (A064152)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Guy A2 (3rd ed., p. 11). Heuristic count #E(x) ~ (1/8) x / log x (OEIS comment, fitted to census through 10^9) strongly suggests infinitude, i.e. positive relative density among primes - which is exactly why new examples carry no information. Erdos's suggested easier variant (per erdosproblems 1059 remark): show infinitude of n in (l!, (l+1)!] whose prime factors all exceed l and with n - k! composite for 1 <= k <= l; still not witness-shaped." }, { "problem": 1060, "oeis": "A327153", "quantity": "A212490(6) - least m such that k*sigma(k) = m has exactly 6 solutions (the record ladder for f(n) = A327153, the count Erdos asked to bound)", "kind": "value_gap", "lower": { "value": "> 5*10^14", "year": 2020 }, "upper": { "value": "7089671638182002688000", "year": 2020 }, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdos problem 1060 proper (Guy B11) asks whether f(n) = #{k : k*sigma(k) = n} is <= n^(o(1/loglog n)), maybe (log n)^O(1) - analytic wall. The machine lane is A212490: known a(1)..a(5) = 6, 336, 333312, 5418319872, 1584858562560 (each ~ products of perfect numbers times 2-powers); every new cell is also evidence that f is unbounded, itself unproven. Adjacent cells are open too, some with SMALLER upper bounds than a(6): Suteu gives a(7) <= 106345074572730040320 (~1.06e20) and bounds for a(8), a(9); Corneth: conditional pushes to 1.1e17 / 1.8e20 for the lower side; Suteu conjecture a(n) <= 2^(n-1) * prod_{k<=n} A000396(k). Corneth's Pari program is linked in the OEIS entry. exact_feasibility graded cell on the strength of the bounded sqrt(m) sieve, but flag: if a(6) sits near the upper bound the collision sort is ~670GB of keys - big-box territory, not laptop." }, { "problem": 1061, "oeis": "A110177", "quantity": "S(x) = number of ordered pairs (a,b) with a+b <= x and sigma(a) + sigma(b) = sigma(a+b): is S(x) ~ c*x for some c > 0?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "IMPORTANT FRESHNESS FLAG: a June 2026 preprint claims to RESOLVE this problem negatively - Eric Li (Trinity College, Cambridge), 'A resolution of Erdos Problem 1061 on the sum-of-divisors function', arXiv:2606.25849 (submitted 2026-06-24), proving S(x)/(x (log x)^R) -> infinity for every R > 0, refuting the ~c*x guess. The preprint is linked from OEIS A110177 but is NOT peer-reviewed, and erdosproblems.com/1061 still shows OPEN as of 2026-07-18 (comment activity pending). Recorded as open per upstream, but the scout should spend NO compute here and watch for the upstream status flip; if Li's proof stands, mark closed. Guy B15." }, { "problem": 1062, "oeis": "A038372", "quantity": "A038372(45) - largest subset of {1,...,45} in which no element divides two other distinct elements (fork-free sets)", "kind": "next_cell", "lower": { "value": "30", "year": null }, "upper": { "value": "31", "year": null }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "a(45) is boxed into {30, 31}; a 45-variable ILP/SAT solves it in seconds, and the whole table (data stops at n=44, no b-file) extends by dozens of exact cells in hours - among the cheapest genuine table records in the hub. The underlying Erdos problem (Guy B24): how large is f(n), and is lim f(n)/n irrational? Interval (n/3, n] gives f(n) >= ceil(2n/3); Lebensold 1976: 0.6725 n <= f(n) <= 0.6736 n for large n (stated as a theorem in formal-conjectures 1062.lean and on the problem page) - tightening THAT density gap is a separate, analytic/LP-flavored lane; new exact cells sharpen the empirical density toward the irrationality question. erdosproblems 1062 has 13 comments with claimed partial results - check before attacking the density constants." }, { "problem": 1063, "oeis": "A389360", "quantity": "A389360(60) = n_60 — least n >= 120 such that n-i divides binomial(n,60) for all but exactly one i in 0<=i<60 (Erdős–Selfridge quantity n_k at the first uncomputed k)", "kind": "next_cell", "lower": { "value": "120", "year": null }, "upper": { "value": "581442729886633902054768000", "year": 2026 }, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "b-file complete and gap-free for k=2..59 (Chai Wah Wu, Oct 2025) — a(60) is the live frontier. Values are erratic: a(57)=218175259280 (~2.2e11), a(59)=4967281530 (~5.0e9); an exhaustive scan to ~1e11-1e12 with modular tricks is days-scale on one box, so exact a(60) is a bounded search (same method as the existing terms). Theoretical: Erdős–Selfridge observed at least one exception index always exists for n>=2k (proof in Monier via [Mo85]); Monier bound n_k <= k! improved by Cambie to k*lcm(2..k-1) <= e^{(1+o(1))k}. Guy UPINT problem B31. OEIS entry created Oct 2025 (Cambie), so freshness risk is low." }, { "problem": 1073, "oeis": "A256519", "quantity": "A(x) = number of composite u <= x with u | n!+1 for some n — conjecture A(x) <= x^{o(1)}", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős question from discussions with Hardy and Subbarao (Guy UPINT A2). Enumeration state (verified from b-file): 719 composites known, largest a(719) = 99795337, i.e. A(~1e8) = 719 — consistent with very slow (x^~0.36 at this range) growth but of course probing nothing asymptotic. Wilson's theorem makes the prime analogue trivial, which is why only composite u count. Extending the enumeration is open-easy compute (sieve over u, scan n < u) but is data-gathering, not a record. Stub statement (Apache/Lean): 'Is it true that A(x) <= x^{o(1)}?'" }, { "problem": 1083, "oeis": "A186704", "quantity": "A186704(14) — minimum number of distinct distances determined by 14 points in the plane (first uncomputed cell of the linked sequence; the hub problem itself asks for f_d(n) asymptotics in fixed dimension d >= 3)", "kind": "next_cell", "lower": { "value": "7", "year": 2012 }, "upper": { "value": "7", "year": 2026 }, "witness_side": "upper", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "CROSSREF CLOSURE AVAILABLE: a(14)=a(15)=a(16)=7 all follow from Wei 2012 (+ EF96/Shinohara08 for j<=5) plus lattice constructions verified this session — OEIS still ends at a(13)=6, so extending A186704 (and A131628, whose a(6)=13 already encodes Wei) is a free librarian win in the #241/#552 pattern. First genuinely open cell: a(17) ∈ {7,8} — verified 17-point patch gives <= 8; equals 7 iff a 17-point 7-distance set exists, i.e. iff g(7) >= 17; best known 7-distance construction has 16 points and the classification of maximum planar 7-distance sets is announced as near-term work (arXiv:2510.09800, Oct 2025) — that cell is 'hard' (EF-conjecture-era case analysis). CAVEAT: hub problem 1083 is the d>=3 generalisation (asymptotics: Erdős 1946 n^{1/d} << f_d(n) << n^{2/d}; Aronov–Pach–Sharir–Tardos 2004; Solymosi–Vu 2008 f_3 >> n^{3/5} after Guth–Katz; conjecture f_d(n) = n^{2/d - o(1)} open) — that part is a wall with no cheap witness; the OEIS link is marked 'possible' and is the planar (problem 89) table." }, { "problem": 1084, "oeis": "A045945", "quantity": "contact-number problem f_d(n) = max pairs at distance exactly 1 among n points pairwise >= 1 apart; linked sequence A045945 = 3n(3n+1) is the closed-form planar record values", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Open content is d >= 3 asymptotics, a constants gap not a table: Erdős claimed c_1,c_2 > 0 with 6n - c_1 n^{2/3} < f_3(n) < 6n - c_2 n^{2/3}; best proven upper bound f_3(n) < 6n - 0.926 n^{2/3} for n >= 2 (Bezdek–Reid 2013); general d: (d-o(1))n <= f_d(n) <= 2^{O(d)} n (lattice lower bound, kissing-number upper bound); survey: Bezdek–Khan 2018 (contact numbers for sphere packings). A potential future records lane: exact maximum contact numbers c(n) for small n in R^3 (sticky-sphere clusters) — would need a new ledger, the hub's linked sequence does not cover it (link marked 'possible'). See also hub #223 for the diameter-1 analogue." }, { "problem": 1085, "oeis": "A186705", "quantity": "A186705(22) = u(22) — maximum number of unit-distance pairs among 22 points in the plane (first open cell of the linked planar table; hub problem is the general-d unit-distance count)", "kind": "value_gap", "lower": { "value": "60", "year": 2024 }, "upper": { "value": "61", "year": 2024 }, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "One-edge gap: 60 <= u(22) <= 61 (AMP Thm 1(b)); AMP's pipeline (forbidden-subgraph graph generation + custom algebraic embedder) obtained exact u(n) for all n <= 21 (u(16..21) = 41,43,46,50,54,57, matching OEIS data) and full enumerations of densest graphs, but left 22..30 as gaps: u(23) in [64,66], u(24) in [68,72], u(25) in [72,78], u(26) in [76,84], u(27) in [81,90] (lower = Hamming(3,3) graph), u(28) in [85,96], u(29) in [89,103], u(30) in [93,110]. Deciding u(22) is finite via their method but exceeded their 2024-25 budget — grade the exact cell as heavy expert compute. FRESHNESS: OEIS entry edited May 2026 confirms AMP is still current for small n; the 2026 asymptotic developments (disproof-of-unit-distance-conjecture remarks arXiv:2605.20695; Sawin explicit lower bound arXiv:2605.20579) concern large-n growth, not these cells. CAVEAT: hub 1085 is the general-d problem — even d >= 4 is essentially closed (Brass 1997 d=4 exact; Swanepoel 2009 even d >= 6 exact; Erdős–Pach 1990 odd d >= 5 up to constants); the open content is d=2 (this table + asymptotics) and d=3 (n^{4/3}loglog n << f_3 << n^{3/2}beta(n)); OEIS link marked 'possible' (it is problem 90's planar table)." }, { "problem": 1095, "oeis": "A003458", "quantity": "A003458(378) = g(378) — Erdős–Selfridge function: least n > 379 such that the least prime factor of binomial(n,378) exceeds 378 (first cell beyond the verified literature frontier k=377)", "kind": "next_cell", "lower": null, "upper": null, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "FRESHNESS CATCH (verified in primary source): the OEIS b-file stops at k=375 (a(375)=12863999653788432184381680413559) but the Sorenson–Sorenson–Webster arXiv v3 addendum (arXiv:1907.08559v3, June 2021, supplementing their ANTS XIV paper, Open Book Ser. 4 (2020) 23) reports g(376)=7778804220120654420924631668091 (one week wall time) and g(377)=5973303871796437264595936954237 (~two weeks) — so a(378) is the true frontier and adding 376-377 to OEIS is a free librarian win. Expected magnitude ~1e31; SSW's sublinear-in-g(k) sieve is the tool (CRT/knapsack-constructed admissible residues + wheel), so exact a(378) is weeks-scale expert compute, and producing a non-minimal upper-bound witness via their residue construction is much cheaper. Asymptotics on the hub problem: k^{1+c} < g(k) <= exp((1+o(1))k) (Ecklund–Erdős–Selfridge 1974); record lower bound g(k) >> exp(c(log k)^2) (Konyagin 1999); ELS 1993 conjecture direction exp(ck/log k); SSW heuristic log g(k) ≍ k/log k; EES conjectured g(k) < lcm(1..k) for large k. See also hub #451, #1094." }, { "problem": 1100, "oeis": "A325864", "quantity": "g(k) = max over squarefree n with omega(n)=k of tau_perp(n), where tau_perp counts coprime consecutive-divisor pairs — exponential growth base", "kind": "value_gap", "lower": { "value": "(sqrt(2)+o(1))^k", "year": 1981 }, "upper": { "value": "(2-c)^k for some absolute c>0", "year": 1981 }, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "The bounds are asymptotic bases with unspecified o(1)/c — recorded as expressions, not numerics; nobody has published a finite-k g(k) table as far as checked. Finite-k values are suprema over infinitely many prime choices but depend only on the realizable order type of the 2^k subset-log-sums, so bounded search per k is plausible for small k (hyperplane-arrangement enumeration); grading exact as unknown. Other parts of hub 1100 are asymptotic and not witnessable: tau_perp(n)/omega(n) -> infinity for almost all n; tau_perp(n) < exp((log n)^{o(1)}); Erdős–Hall 1978 proved max_{n exp((loglog x)^{2-eps}). Trivial: tau_perp(n) >= omega(n), with equality infinitely often. OEIS link A325864 is marked 'possible' and counts sum-distinct subsets of {1..n} (b-file 0..150) — related machinery, not a g(k) ledger. Page flags unincorporated partial-result claims in comments — check before claiming novelty." }, { "problem": 1103, "oeis": "A392164", "quantity": "Growth rate of an infinite sequence A of integers such that every element of A+A is squarefree (how slowly can a_j grow?)", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Asymptotic-only problem (erdosproblems.com marks it not resolvable by finite computation). Known: every such sequence satisfies a_j > 0.24·j^(4/3) for all j, and a squarefree-termed example exists with a_j < exp(5j/log j) for large j (van Doorn–Tao, arXiv:2512.01087, 2025); Konyagin (Izv. RAN 2004) had earlier a_j >> j^(15/11-o(1)). The finite analogue is problem #1109 (same OEIS A392164), which IS machine-actionable — see that entry. Upper bounds for f(N) in #1109 imply lower bounds here." }, { "problem": 1104, "oeis": "A292528", "quantity": "A292528(6) — minimum number of vertices in a triangle-free graph with chromatic number 6 (first unknown cell of the sequence 1,2,5,11,22,?)", "kind": "value_gap", "lower": { "value": "32", "year": 2020 }, "upper": { "value": "40", "year": 2020 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Bounds 32 <= a(6) <= 40 have stood since 2017; OEIS entry is actively curated (edits Jan 2026 by Greathouse added asymptotics without touching them), and 2026 web check found nothing newer. Witness graded hard because the 33..39-vertex window was already attacked by the domain expert with dedicated triangle-free generators (triangle-free Ramsey-style enumeration); a lucky heuristic/tabu search hit is conceivable but ungraded optimism is banned. Lower-bound side (33) needs exhaustive nonexistence — a Sigma2-type constraint (for-all colorings), not naturally DRAT. Known values: a(4)=11 unique (Chvátal 1974, Grötzsch graph), a(5)=22 with exactly 80 extremal graphs (Jensen–Royle 1995). Asymptotics for the problem's f(n) (max chromatic number of triangle-free graph on n vertices): between (1-o(1)) and (2+o(1)) times sqrt(n/log n) (upper: Davies–Illingworth arXiv:2107.12288; lower: Hefty–Horn–King–Pfender 2025, per erdosproblems.com/1104). Mycielski recursion gives a(n) <= 2a(n-1)+1, so a(6) settles the next rung of a whole ladder." }, { "problem": 1106, "oeis": "A194260", "quantity": "Conjecture F(n) > n for all n > 115, where F(n) = A194259(n) = number of distinct primes dividing p(1)·p(2)···p(n) (p = partition function); A194260(n) = F(n) - n", "kind": "verified_range", "lower": { "value": "10000", "year": null }, "upper": null, "witness_side": "lower", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Problem asks (a) does F(n) -> infinity (answered: yes, follows from Tijdeman-type results as noted by Schinzel; details in Erdős–Ivić 1990) and (b) is F(n) > n for large n — open. Proven: F(n) >> log n (Schinzel–Wirsing, Proc. Indian Acad. Sci. 1987); every prime divides some p(n) (Ono 2000). Empirically F(n) - n is positive for all 115 < n <= 10000 and reaches 18312 at n = 10000 (b-file, re-verified this session). Extension graded plausible, not open-easy: p(10000) has ~106 digits and each new term needs a full factorization (ECM + GNFS-class effort on stubborn cofactors), days per stubborn term. Asymptotic question itself is a wall — erdosproblems.com marks it not resolvable by finite computation." }, { "problem": 1107, "oeis": "A392342", "quantity": "A392342 — integers not the sum of at most 4 cubefull numbers (the r=3 case of: every large integer is a sum of at most r+1 r-powerful numbers). Conjecturally finite with largest term 2039; record = verified search bound", "kind": "verified_range", "lower": { "value": "1e9", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Erdős–Ivić (Oberwolfach 1986). r=2 case SOLVED: Heath-Brown 1988 proved every large integer is a sum of at most 3 powerful numbers; A056828 (non-sums of <= 3 powerful) is conjecturally complete at {7,15,23,87,111,119} (Mollin–Walsh 1986), checked to 4x10^7 (Jobling 2001). Open for every r >= 3. r=3 sieve frontier is 10^9 (Cleaver, Feb 2026) — pushing to 10^11+ is a straightforward table-of-cubefull + 4-sum sieve, hours-days on one box, and each new decade strengthens the finiteness evidence. Fresh sibling lane: the r=4 case A392343 (non-sums of <= 5 four-full numbers) has NO recorded finiteness frontier yet — computing its presumed-final term and a verified bound would be a first, not an improvement. The full all-large-n statement is a wall (not finitely decidable)." }, { "problem": 1107, "oeis": "A056828", "quantity": "Mollin-Walsh completeness frontier (A056828) — verified no new n fails to be a sum of <=3 powerful numbers", "kind": "verified_range", "lower": { "value": "10^6", "year": 2026 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "record_in_progress", "confidence": "C2", "note": "PUBLIC BOUND: no seventh exception through 10^6, replayable in-repo. PRIVATE OPERATIONAL RESULT: a private verified-up-to-N run reported 10^10 with A118896 and sampled-witness cross-checks, but no canonical receipt was found in the public package during the 2026-07-21 audit; it is quarantined from promotion until packaged. The full conjecture is a wall." }, { "problem": 1108, "oeis": "A115645", "quantity": "A115645(27) — next powerful number (min prime exponent >= 2) expressible as a sum of distinct factorials; 26 terms known, largest = 1401602635449 = 1183893^2", "kind": "verified_range", "lower": { "value": "1e18", "year": 2024 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Erdős at Oberwolfach 1988, motivated by a question of Mahler. Range push is genuinely cheap: all subset sums of {1!..21!} number only 2^21 ≈ 2·10^6 with max ~5·10^19, and subsets of {1!..25!} are 3.4·10^7 with max ~1.6·10^25 — factoring that many <=84-bit numbers is days on one box, so 10^18 -> 10^24 is within reach (grade open-easy for the verified-range record; an actual 27th term is conjecturally unlikely). The squares-only subcase A025494 is already far deeper: 16 squares known (largest 1401602635449), and any further square must exceed 48! ≈ 1.24·10^61 (Schoenfield 2006, via A014597) — squares need no factorization, hence the deeper range. Theory: Brindza–Erdős 1991 proved for fixed r that powerful sums of r factorials have bounded largest factorial. Convention warning: A115645 treats 0!,1! as equal; A051761 (perfect powers variant, 21 terms) treats them as distinct. Finiteness question itself is a wall." }, { "problem": 1109, "oeis": "A392164", "quantity": "A392165(40) — smallest N with f(N) >= 40, where f(N) = A392164(N) = size of the largest S ⊆ {1..N} such that every element of S+S (pairwise sums, including 2a) is squarefree. Table exact through N = 1103 (f(1103) = 39)", "kind": "next_cell", "lower": { "value": "39 (f exactly computed through N = 1103; 39th record index = 1103)", "year": 2026 }, "upper": null, "witness_side": "lower", "witness_feasibility": "open-easy", "status": "open", "confidence": "C3", "note": "Erdős–Sárközy 1987 (Acta Math. Hungar.): log N << f(N) << N^(3/4) log N; best known asymptotics are Konyagin 2004: loglog N (log N)^2 << f(N) << N^(11/15+o(1)) (OEIS formula lines + erdosproblems.com/1109; van Doorn–Tao arXiv:2512.01087 revisits the problem family). Erdős–Sárközy guessed the lower bound is nearer the truth; the problem asks if f(N) <= N^o(1). Machine lane: build the compatibility graph on {1..N} (vertex a admissible iff 2a squarefree; edge iff a+a' squarefree) and run an exact max-clique solver (cliquer-class) — N in the low thousands is hours of compute, extending both A392164 (only 98 terms listed, no b-file) and A392165 (39 record indices, keyword 'more'). A392165 record indices are all odd squarefree (subsequence of A056911, per Wu). This finite table is also the machine surrogate for infinite-version problem #1103." }, { "problem": 1113, "oeis": "A076336", "quantity": "Existence of a Sierpiński number (odd k with k·2^n+1 composite for all n >= 1) that has NO finite covering set of primes", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Erdős–Graham 1980 (p.27), also Guy UPINT F13. Sierpiński 1960: infinitely many Sierpiński numbers exist (positive density, via covering systems); smallest is believed to be 78557 (Selfridge). Concrete evidence for YES: m = 734110615000775^4 is a proven Sierpiński number (Izotov, Fibonacci Q. 1995; detailed by Filaseta–Finch–Kozek 2008) whose compositeness for n ≡ 2 mod 4 comes from an Aurifeuillian factorization rather than a covering prime — believed to have no finite covering set. Filaseta–Finch–Kozek revised conjecture: every Sierpiński number is a perfect power or has a finite covering set. A negative answer would imply infinitely many Fermat primes. Adjacent (different-problem) machine lane, noted for the scout: proving 78557 smallest = Seventeen-or-Bust residue; a Proth prime k·2^n+1 for a remaining k is a cheaply verifiable world-class witness, but it belongs to the 'smallest Sierpiński number' question, not #1113." }, { "problem": 1135, "oeis": "A008908", "quantity": "Collatz (3x+1) conjecture — every positive integer reaches 1; machine-facing quantity: the exhaustively verified convergence bound", "kind": "verified_range", "lower": { "value": "2^71", "year": 2025 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "Hub status: wall — correctly so. Not originally Erdős's problem (Collatz, pre-1952); the $500 figure traces to a 1983 Erdős–Graham conversation reported by Lagarias, not a formal prize offer; Erdős called such problems ones mathematics 'may not be ready for' (Guy UPINT E16 / Lagarias 2016 notes, paraphrased from erdosproblems.com/1135). Theory rungs: at least N^0.84 of integers below N eventually reach 1 (Krasikov–Lagarias 2002); almost-all logarithmic-density convergence (Tao 2019, not re-verified this session — flagged as context not a recorded bound). GPU verification throughput exists (Barina's total CPU->GPU speedup reported at ~1335x), but a fleet-scale re-run to claim a new record without novel algorithmic leverage is not a rational use of the farm. Recommend the scout treat this as a monitor-only board." }, { "problem": 1137, "oeis": "A005250", "quantity": "Does max_{n 0 as x -> infinity, where d_n = p_{n+1} - p_n (are two consecutive record-sized prime gaps asymptotically forbidden)?", "kind": "not_gap_shaped", "lower": null, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "From Vaughan's problem collection attribution [Va99, 1.2] per erdosproblems.com/1137 (page carries no partial results — the remarks section is empty, which is itself signal: nothing to be scooped by). Data context: A005250 (record prime gaps) b-file now has 85 maximal gaps, largest listed 1854 (b-file extended by Kehrig, fetched and read this session — exhaustive frontier has moved past the long-standing 2^64 era); A083550 = d_n·d_{n-1} has a 10000-term b-file. Cramér-type heuristics make consecutive near-maximal gaps doubly exponentially rare, favoring YES, but nothing is proven. A candidate scout lane — tabulating record values of A083550 and their positions from extended gap tables — is cheap and publishable as OEIS data, but honestly bounds nothing about this problem; graded not-gap-shaped." }, { "problem": 1142, "oeis": "A039669", "quantity": "A039669 — integers n > 2 with n - 2^k prime for every k >= 1 with 2^k < n; known terms {4, 7, 15, 21, 45, 75, 105}, conjecturally complete (Erdős). Record = verified completeness bound", "kind": "verified_range", "lower": { "value": "2^120", "year": 2011 }, "upper": null, "witness_side": "lower", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "Erdős 1950 (Summa Bras. Math.); Guy UPINT A19; [Va99, 1.7]. Erdős conjectured the seven known terms are all (OEIS comment), so the new-term witness likely doesn't exist; the honest record is the completeness bound, and 2^120 (2011, unpublished-in-journal OEIS comment by Alekseyev) already reflects clever covering/sieve work — beating it is graded hard, not because verification is hard but because the low-hanging sieve range is spent. Vaughan 1973 proved the count of such n <= N is at most N·exp(-c·(logloglog N / loglog N)·log N). Erdős's stronger related conjecture (see problem #236): the number of k with n - 2^k prime is o(log n). Curiosity preserved from OEIS: Mientka–Weitzenkamp's 1969 claim of 9 terms below 20000 presumably counted 1 and 2 (Israel 2015 comment). Infinitude/completeness question itself is a wall." }, { "problem": 1160, "oeis": "A000001", "quantity": "Verification frontier m_max for the conjecture: if n <= 2^m then g(n) <= g(2^m), where g(n) = A000001(n) = number of groups of order n (attributed variously to Erdos and Graham Higman; Question 22.16 in Blackburn-Neumann-Venkataraman 2007)", "kind": "verified_range", "lower": { "value": "2048", "year": 2021 }, "upper": null, "witness_side": "none", "witness_feasibility": "none", "status": "open", "confidence": "C3", "note": "erdosproblems.com/1160 marks this open and 'cannot be resolved with a finite computation'. Known partial result (paraphrased from the page's remarks): Pantelidakis [Pa03, 2003] proved the conjecture when n is odd and m >= 3619. BNV07 Question 22.18 proposes the stronger conjecture sum_{n<2^m} g(n) <= g(2^m) for all sufficiently large m (perhaps m >= 7); from the b-file, sum_{n<2^m} g(n) <= g(2^m) FAILS for small m (e.g. m=4: sum=35 > 14) so 'sufficiently large' is essential there. Extending the verified range to m=12 requires g(4096) — far past the enumeration frontier (g(2048) already unknown; only the class-2 lower bound is known). Scout action: none realistic; watch for new Small Groups Library / gnu(n) computations (Konovalov's crowdsourced gnu database on GitHub tracks n < 50000)." }, { "problem": 1204, "oeis": "A008407", "quantity": "A008407(343) = H(343) — minimal diameter a_k - a_1 of an admissible 343-tuple (Elliott-Erdos A(k), Erdos 1980 p.108); k=343 is the first k where the minimal diameter is unconfirmed (exhaustively confirmed through k=342)", "kind": "next_cell", "lower": { "value": "2334", "year": 2013 }, "upper": { "value": "2340", "year": 2009 }, "witness_side": "upper", "witness_feasibility": "hard", "status": "open", "confidence": "C3", "note": "The Erdos/Elliott problem proper is asymptotic and not finitely resolvable: known (1/2+o(1))k log k <= A(k) <= (1+o(1))k log k (lower Elliott 1965, rediscovered by Polymath8; upper attributed to Davenport, via the k smallest primes > k; refinements Hensley-Richards 1973); the open question is whether A(k) ~ k log k. This entry is its machine-actionable shadow cell. Only slack values {2334, 2336, 2338} remain (diameter is even). Feasibility calibration: Engelsma's cell-by-cell exhaustive assembler search (years) set the table, and the 2013-14 Polymath8 tuple toolchain (greedy/Schinzel/ILP; improved Engelsma at ~1927 values of k) did NOT improve k=343 — so an improving witness is graded hard, though a focused modern SAT/ILP attack on widths 2335-2339 has never been tried specifically. Nonexistence side is pigeonhole-like (cardinality 343-of-~2339 + per-prime coverage), so DRAT certificates may blow up; exact closure realistically means Engelsma-style exhaustion just past his 2009 frontier. Related sequences: A023193 = rho*(n) (max admissible tuple in interval of length n; Engelsma: rho*(3159) >= 447 > pi(3159) = 446, so the prime k-tuples conjecture contradicts pi(x+y) <= pi(x)+pi(y) — see problem 855); A135311 = greedy admissible sequence (Erdos also asks its growth; terms 1..10000 in OEIS); B(k) = min-mean variant with expected B(k) ~ (1/2)k log k." }, { "problem": 1208, "oeis": "A193838", "quantity": "A193838(15) — smallest k such that 15 points of the k x k lattice grid have pairwise distinct mutual distances (the Erdos-Guy 1970 finite grid shadow of problem 1208's asymptotic quantity F_d(n)); first uncomputed cell, a(14)=18 exact", "kind": "next_cell", "lower": { "value": "18", "year": 2022 }, "upper": { "value": "20", "year": 2022 }, "witness_side": "upper", "witness_feasibility": "plausible", "status": "open", "confidence": "C3", "note": "Erdos problem 1208 proper (Er57b; Er80 p.110) asks for the asymptotics of F_d(n) = largest subset with pairwise distinct distances guaranteed in ANY n points in R^d — not finitely resolvable. Known (paraphrased from erdosproblems.com/1208 remarks): n^{1/3}/(log n)^{1/3} << F_2(n) << n^{1/2}/(log n)^{1/4} (lower Charalambides 2013 [Ch13]; upper from the integer grid, which has << n/sqrt(log n) distinct distances); for d >= 3, Thiele 1995 gave F_d(n) >> n^{1/(3d-2)}, improved by Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky 2015; F_1(n) ~ n^{1/2} (Komlos-Sulyok-Szemeredi 1975, see problem 530). Grid version (this OEIS pair): Erdos-Guy 1970 showed the max distinct-distance subset of the d-dim grid with n points is >= n^{2/(3d)-o(1)} and <= n^{1/d}; Lefmann-Thiele 1995 improved the lower bound to >> n^{2/(3d)}. Cell history: a(13)=16 Dobbelaere 2020, a(14)=18 Cariboni 2022 — exact values fell to exhaustive DFS with distance-set pruning, so the a(15) decision at k=19 is the next ring of the same bounded search (witness may not exist if a(15)=20). Known upper-bound ladder beyond: a(16) <= 22, a(17) <= 24, a(18) <= 27 (Cariboni 2022); a(19..26) <= 33,36,37,40,43,46,49 (Kamenetsky 2021, Zimmermann contest). Dual cell A271490(18) >= 14 (from the a(14)=18 witness), exact value open; A271490 entry carries keyword 'changed' with a Jul 2026 edit but terms still end at a(17)=13 (b-file checked this session)." } ], "graph": { "nodes": { "crater_node": 37, "evidence_claim": 15, "evidence_container": 28, "external_claim": 91, "incident": 26, "move": 7, "msc": 17, "oeis_seq": 403, "problem": 1217, "surface": 319, "tag": 41, "wall": 214 }, "edges": { "applies": 140, "cascades_to": 29, "claims_about": 91, "classified": 379, "evidenced_by": 11, "has_surface": 319, "learned_from": 5, "measures": 223, "references": 479, "retracts": 17, "same_family": 114, "tagged": 1979, "trap": 7, "walled": 214 }, "tiers": { "cited-implication": 29, "deterministic-rule": 3950, "validator-enforced": 28 } }, "principles": [ { "title": "A claim never sets a status", "body": "Statuses come from the upstream index and from replayable evidence — external claims are verification targets, not results." }, { "title": "Walls render before temptation", "body": "Dead paths and priced traps are published as loudly as targets, so compute is not wasted re-walking closed ground." }, { "title": "The unit of progress is the bracket", "body": "Bounded quantities carry [lower, upper] brackets and a machine-checkable witness shape; confidence is computed from evidence, never asserted." }, { "title": "No prize claim by finite computation", "body": "Almost every Erdős prize attaches to an asymptotic statement. ProjectForty2 reports finite certificates, records, and walls — never a headline solution." } ] }