I'm Patrick — a pretty regular dude, not a mathematician. Every day, my AI partner and I pick one of Paul Erdős's still-unsolved problems and go after it, live. Some days we crack one. Most days we don't. No editing, no do-overs — this is the actual record. Below is the complete ledger: every problem we've touched, the full write-ups and the raw working reports alike.
Owings's 1974 two-colour sumset question was answered affirmatively eight days ago (Huang–Lian–Shao–Xiao–Xu–Zhang, arXiv:2607.17333). The live page still said OPEN. We found the preprint, audited the proof (digest-pinned v2, no gap modulo the named Stone–Čech inputs), and independently certified R_Σ(1..3)=(1,7,23). The theorem is theirs; the catch and the audit are ours.
Read the report ↗The 3/7 upper bound is NOT new to us: a public Star Fleet Math bundle (14 Jul 2026, starfleetmath.com) states it with a Lean 4 formalization of the same four-projection proof. We independently reconstructed the finite argument, audited the Lean sources (no sorries; full Mathlib rebuild not run), and tightened the rounding trivially — so no priority claim, matching the report's own stance. The genuinely new bit is the lower-bound transfer from Kominers 2607.10431 → (1/e−o(1))log n, which the live page omits.
Read the report ↗Bae's arXiv:2604.23784 (April; claims #684) rests on Lemma 18, which is false: a counterexample built in the paper's own definitions, deterministic checker green, the A_p definition and θ-condition matched verbatim against the source. The unbounded-limsup corollary falls with it. A note to the tracker / the author is the next step — Patrick's call.
Read the report ↗Exact exhaustive certificates now give f(1..7)=1,2,2,3,3,4,4 and F(1..7)=1,2,2,3,4,5,7; both asymptotic Erdős-Ulam questions remain open.
PARTIALLive-page status is OPEN with 0 formally listed claimed proofs, but VPendyala is marked “Currently working on this problem” and the live discussion contains Venkata Pendyala’s 18 June 2026 claim of a
SKIPPEDExact exhaustive computation extends A389182 with \(F(101)=\cdots=F(106)=16\), \(F(107)=\cdots=F(116)=17\), \(F(117)=\cdots=F(134)=18\), \(F(135)=\cdots=F(152)=19\), and \(F(153)=20\); the asymptotic
PARTIALexact s(k,n) for all n<=37 plus a proof that Pikhurko's 1.863949... bound is optimal within every mode-by-mode Fourier/triangle-inequality certificate; the asymptotic problem remains open.
PARTIALProved the exact ceiling–Golomb-ruler equivalence and, with dependency-free exhaustive searches and an independent C++ audit, determined f(N) exactly for every N<=120; independently audited the unrefe
PARTIALExhaustive maximum-degree CNFs and complete canonical censuses give h(18)=5 at computer-assisted solver-assertion level; the asymptotic problem remains open.
PROVEDProblem #773 asks whether the largest Sidon subset of the first N positive squares has size N^(1-o(1)); that asymptotic problem remains open. Conditional on the full published exact OEIS A390813 prefix through N=68, frozen exhaustive searches prove the next three values: a(69)=32, a(70)=33, and a(71)=33. The new upper search at 71 covers seven disjoint cubes and 114,296,406 recursive states; the size-33 lower witness is a direct integer certificate. An independent definition-level audit reconstructs every collision support and checks the witness, cube cover, and counters. This extends a finite table, not the exponent in Erdős's question.
PARTIALFULL WRITE-UPExact finite table through 15, plus \(f(17)=10,f(24)=14,f(26)=15,f(28)=16\), with explicit integer blocks and dependency-free checkers; the asymptotic constant remains \(9/17\le c_*\le4/7\).
PARTIAL(d) Exact exhaustive certificates now give \(f_3(N)\) through \(N=125\), with \(f_3(N)=14\) for \(101\leq N\leq107\), 15 for \(108\leq N\leq124\), and 16 at \(N=125\); the asymptotic problem remains o
PARTIALproved an elementary finite-to-infinite extension and exact two-row thresholds through L_40=509, including L_11,...,L_40=101,107,127,131,149,167,173,191,197,211,241,251,257,271,281,307,313,347,361,367
PARTIALExact strict-pair values F(1),...,F(61) are exhaustively verified with standalone checkers and independent SAT controls; the asymptotic constant and higher-r bound remain open, with the missing unifor
PARTIALExact exhaustive certificates prove f(N)=14,16,18,21,23,24,26,28,32,32,34,38,39,41,44,45,48,52,52 for N=8,...,26 in Koizumi's equivalent lattice model; an all-N construction also rules out any one O(N
PARTIALExact dynamics produce arbitrarily long constant nonzero gap transients of either sign (then an absorbing zero tail), refuting uniform small-gap fixed-window strategies, while finite total negative ga
PARTIALA complete computer-assisted census proves Tuza's inequality for every simple graph on at most ten vertices; the unrestricted Erdős problem remains open.
PARTIALFor k(3,4), every order-21 witness is reduced to six canonical root profiles, all eight cyclic order-20 witnesses are exactly nonextendable unchanged, and a checked order-21 near witness has exactly o
PARTIALExact independent computations now give \(M_k(9)=(0,18,21,27,28,30,32,36)\), extending the complete frontier through nine vertices; the asymptotic problem remains open at the same triangle-removal sca
PARTIALAn exact formula counts p-subgroups of S_n by order when every nontrivial orbit has size p. One exact order already attains the known n^2/16 leading exponent, with an explicit n log n term, and the same restricted family has a discrete order limit law. This refines a known construction without solving the global problem.
PARTIALFULL WRITE-UPAn exact reconstruction gives 49,910,538,480 groups across all orders below 2,048. A published exact subclass count of 1,774,274,116,992,170 at order 2,048 already exceeds that total, certifying the stronger conjecture at m=11. The general problem remains open.
PARTIALFULL WRITE-UPBrutman and Toledano computed the four-node optimum numerically in 1997. We give an independently checkable exact global certificate: the unique nodes are {-1,-sqrt(y),sqrt(y),1}, where 21y^3+9y^2+3y-1=0, and the minimum is algebraic of degree three. This finite result does not settle the asymptotic problem, and no priority claim is made for the exact formulas.
PARTIALFULL WRITE-UPA July 2026 preprint strengthens the known affirmative limsup result to a finite-order class-B entire function. Separately, a polynomial whose nonzero exponents span d-m has at most d-m maximum-modulus points on every circle, sharply. The liminf question remains open.
PARTIALFULL WRITE-UPThe asymptotic problem remains open. A second exact computation adds ten transition indices after 1500 and proves f(N) exactly through N=2000. A custom verifier and official PMC agree on all 201 new eligible last-vertex decisions.
PARTIALFULL WRITE-UPComplete factorization gives F(115)=115 and F(116)=117. From there, one fresh proven-prime divisor of each partition number p(n) certifies F(n)>=n+1 for every 116<=n<=10,100. The 100 new steps extend the published finite frontier; the eventual inequality remains open.
PARTIALFULL WRITE-UPErdős and Surányi's 1959 paper defined a constant cₙ — how long an interval, as a multiple of max(A), guarantees n picks whose product is divisible by ∏A — proved c₂=1 and c₃=√2, and reported no bounds in general for 67 years. Rounds 1–7 pinned c₄ through c₁₀ at 2 and proved cₙ ≥ 2 for every n ≥ 4. Round 8 proves c₁₁=2. A maximum-defect Hall reduction leaves finite eleven-edge two-, three-, and four-defect kernels. The new difficulty is placing outside-edge representatives and repair multiples jointly: multi-repair obstructions collapse by short progression lemmas, while the sole singleton obstruction is removed by a gcd/lcm exchange. Rado's theorem handles K₃,₄−e, and robust support blocks survive the aligned injection's two-adic matching while avoiding the midpoint. This determines cₙ through n=11 and improves the general bound to cₙ≤2n/11+14/11. Independent hostile audits cleared the integrated proof. The famous g(n)≤2n conjecture stays open and untouched.
LIVEFULL WRITE-UPProblem #617 asks whether every r-coloring of K_(r^2+1) has an (r+1)-vertex clique missing a color. That remains open beyond the known cases r=3,4. We prove an exact auxiliary theorem for every r>=2: the fewest edges in an (r^2+1)-vertex graph with neither a clique nor an independent set of size r+1 is (r^3-r^2+4r-2)/2. Thus every color class in a hypothetical counterexample has at least 59 edges at r=5. The five bounds total 295 of 325 edges, leaving a real 30-edge compatibility gap, so this is a structural frontier rather than a solve.
PARTIALFULL WRITE-UPAn unreviewed public candidate proof gives R(C₄,K₁,₃₉)=46, closing the first bracketed entry in the finite table. Our hostile audit reconstructed the 45-vertex lower witness, the universal 46-vertex reduction, the exact spectral moments, and a rational dual certificate, and found the finite proof valid without relying on the candidate's uncertified SAT logs. This is an independently audited candidate result, not yet peer reviewed and not a solution of the asymptotic Erdős problem.
PARTIALFULL WRITE-UPProblem #295 asks whether the shortest unit-fraction expansion above a denominator cutoff eventually exceeds (e−1)N by an unbounded amount. That asymptotic problem remains open. Round 1 closed k(17)=32. Round 2 now proves the rigorous computational bounds 33≤k(18)≤35. The harmonic maximum excludes at most 30 terms; exact GMP and Fraction recursions independently execute the same proved exhaustive search and exclude 31 and 32. A transparent four-for-one unit-fraction surgery on the k(17) identity supplies 35 terms beginning at 18. Full reproduction, sanitizer, serial, concurrency, and hostile source audits passed. Whether k(18) is 33, 34, or 35 remains open, as does Erdős's asymptotic question.
PARTIALFULL WRITE-UPAn exhaustive scan from zero finds the next record gap between sums of two squares at 133,858,454,292..133,858,454,404. Its difference is 112, so 111 consecutive interior integers are nonrepresentable. Independent scans and a direct factor certificate prove it is the first new record and remains maximal through 200 billion; the asymptotic problem remains open.
PARTIALFULL WRITE-UPThe published diameter-22,270 integral heptagon has no distinct eighth point whose seven distances to it are positive integers at most 96,000. Three exact exhaustive traversals and a big-integer metric audit certify the bounded nonextension. This concerns one fixed heptagon and gives no lower bound for arbitrary eight-point sets.
PARTIALFULL WRITE-UPThree exact enumerations give theta_4(13)=949,812,334 and theta_4(14)=9,471,574,188. The known table through n=12 is independently reproduced. This is a finite counting extension only; the singly infinite permutation problem remains open.
PARTIALFULL WRITE-UPA preprint posted on 10 August proves that adjoining any point outside the affine hull of a finite Euclidean Ramsey set preserves the Ramsey property. This resolves Ivan-Leader-Walters Conjecture 8 and extends their earlier subsoluble-base result to arbitrary Ramsey bases. An independent proof audit found no gap; the full characterization in Problem 174 remains open.
PARTIALFULL WRITE-UPA reproducible floating-MIP computation extends the finite-value table for corner-free subsets of the first 3-smooth numbers from prefix 5,000 through 5,020, adding twelve OEIS A004059 rows. Two SCIP formulations agree; no proof-carrying certificate was obtained, and the density and irrationality questions remain open.
PARTIALFULL WRITE-UPDoes a maximal Sidon subset of {1,...,N} exist with size O(N^(1/3))? The asymptotic problem remains open, and Rounds 1–2 explain why the extra log in Ruzsa's construction is structural for random lifts. Rounds 3–5 close the next exact finite frontier: a(66)=a(67)=6, a(n)=7 for every 68≤n≤101, and a(n)=8 for every 102≤n≤136. Complete enumerations cover normalized Sidon sets through seven marks, while explicit maximal sets certify every upper bound. This extends OEIS A382397 from 65 through 136 without changing the parent problem's asymptotic record.
LIVEFULL WRITE-UPA July 2026 strong-clique bound, specialized at maximum degree 4, proves the exact value 20; the C5 two-blowup attains it. This is a literature-derived partial. The strong chromatic maximum remains between 20 and 21, so Problem 149 is still open.
PARTIALFULL WRITE-UPClemen, Dumitrescu, and Liu proved that every convexly positioned planar n-point set, n>=5, has a non-diameter distance occurring at most n times. Their proof combines the Altman-Fishburn equality classification with a short pair count. This is a published partial; the arbitrary-set and growth questions remain open.
PARTIALFULL WRITE-UPAn explicit subgraph of the 9-dimensional cube has 1,531 edges and no 4-cycle, proving ex(Q9,C4)>=1,531. Exhaustive face and common-neighbour checks certify the graph. It improves our earlier 1,506-edge certificate by 25 edges and the identified external 1,505-edge certificate by 26; this remains a finite lower bound.
PARTIALFULL WRITE-UP[a+d] Explicitly proved \(f_d(n)\ge\binom{d+1}{\binom n2-1}+1\), certified \(f_1(4)=7\), and reduced the fixed-\(n\) subexponential question equivalently to distinct-distance subsets on \((d-1)\)-sphe
PARTIALLive page status OPEN (0 claimed proofs), but “Currently working on this problem” lists dumbprime; stopped before attempting mathematics to avoid colliding with the current worker.
SKIPPEDLive page status OPEN; it lists 1 claimed proof—a partial proof claimed by Ákos Dúcz and Dániel Varga, submitted 2026-07-29, asserting a finite planar unit-distance graph with independence ratio stric
SKIPPEDThe authoritative live page is OPEN with 0 claimed proofs, but lists Hotdingus as currently working on problem #1049, so the required no-collision rule applies.
SKIPPEDLive page status OPEN (accessed 2026-07-31), with 0 claimed proofs, but “Currently working on this problem: epistemologist”; mandatory non-collision stop.
SKIPPEDThe live page is FALSIFIABLE/open with 0 claimed proofs, but lists “Currently working on this problem: ogroth” (accessed 2026-07-31), so the mandatory no-collision rule applies.
SKIPPEDReduced the live question exactly to compact capacity-one sets, proved an exact Chebyshev lemniscate-area formula with sharp constant \(C_*=1.579155688354179\ldots\), and isolated critical-value separ
PARTIALThe live page status is OPEN (0 formal claimed proofs; “Currently working” is None), but its comments contain a claimed and independently endorsed proof that \(\rho(f)\ge(\log 2)/n\) for every admissi
SKIPPEDThe live page is OPEN but lists 2 claimed proofs and names naprienko and danrobinson as “Currently working on this problem” (accessed 2026-07-31), so the mandatory no-duplication/no-collision rule for
SKIPPEDThe live page is OPEN (accessed 2026-07-31) and shows no current worker, but it lists 2 full proof claims—by Shouqiao Wang (submitted 2026-07-21) and Sangyoon Kwon (submitted 2026-07-23)—so the mandat
SKIPPEDThe live page is OPEN (0 formally logged claimed proofs; Currently working: None), but its discussion has a 27 Apr 2026 comment by Przemek Chojecki saying the problem “was claimed” in the linked prepr
SKIPPEDLIVE PAGE STATUS OPEN (0 formal claimed proofs; no current-worker or interested markers), but its 21–22 April 2026 discussion contains a claimed solution—arXiv:2604.18535 is said to give negative answ
SKIPPEDThe authoritative live page is OPEN but lists “1 claimed proof for this problem”—a full proof claimed by RayYoung, Keheng Zhu, and Yanping Luo, submitted 2026-07-15—so the mandatory no-duplication gat
SKIPPEDLive page status is OPEN, but it lists “1 claimed proof for this problem” (a full proof claimed by Eric Hou, submitted 2026-07-21); “Currently working on this problem” is None.
SKIPPEDproved a rigorous upper-constant improvement to \(1/(2\ln2)=0.7213475\ldots\) in base-2 normalization and exhaustively computed \(f(3),\ldots,f(9)=1,4,11,23,44,77,123\), but the limit and its value re
PARTIALLive page status is OPEN with 0 claimed proofs, but it lists “Currently working on this problem: SamKorsky” (accessed 2026-07-31); the mandatory collision rule therefore forbids an attempt.
SKIPPEDLive page status OPEN (last edited 11 May 2026); “Currently working on this problem” lists ligma and SamKorsky, so the no-collision rule requires stopping.
SKIPPEDLive page status is OPEN - $250, but it lists 2 full proof claims (QuietMethod and Liam Price) and names TheTorturedPoetsDepartment as currently working, so the mandatory no-duplication/no-collision r
SKIPPEDproved a strict Melchior/Bojanowski upper-bound improvement, exact grid formulas with a checker, and uniform lower bounds; the constants and limit existence for k>=4 remain open.
PARTIALThe live page status is “OPEN - $25” but it lists “2 claimed proofs for this problem” (with “Currently working on this problem: None”), so Step 0 requires no attempt.
SKIPPEDLive page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists Ritvik_Nayak (accessed 2026-07-31), so the mandatory no-collision rule requires stopping.
SKIPPEDExact computer-assisted determination \(g(1),\ldots,g(9)=1,2,2,3,4,4,4,5,5\), with rational witnesses and a real nonrepresentability proof at \(n=8\); the verified asymptotic gap \(\Omega(\sqrt{n\log
PARTIALLive page status is `OPEN - $100`, but “Currently working on this problem” lists `MrLaine` (accessed 2026-07-31); per the collision rule, no attempt was made.
SKIPPEDThe live page status is OPEN, but it lists Sam_Petkov as “Currently working on this problem,” so an attempt would collide with a current worker.
SKIPPEDProved a uniform injective layer reduction, closed forms for the top three axis-support coefficients for every n, and independently certified exact n<=10 polynomials and strict two-sided bounds for di
PARTIALThe authoritative live page is OPEN with 0 claimed proofs, but it lists msawhney as “Currently working on this problem” (accessed 2026-07-31), so no mathematical attempt was made.
SKIPPEDLive page status is OPEN (0 formal claimed proofs; “Currently working” marker: None), but its April 2026 comments explicitly claim possible/complete solutions and state that Brayden Letwin is actively
SKIPPEDLive page status OPEN, but it lists 1 claimed proof (a full proof claimed by Rob Sneiderman, submitted 2026-07-21) and identifies lof310 as currently working on the problem.
SKIPPEDExplicitly constructed and verified an infinite-order, non-Fejér Fabry-gap entire function that assumes every value infinitely often; the universal problem reduces exactly to excluding zero-density Ta
PARTIALThe live page status is OPEN, but its discussion identifies current workers: on 5 June 2026 Przemek Chojecki said that he and Yuta were jointly preparing a paper collecting all arguments for Problem #
SKIPPEDAn explicit He–Tang-family entire function and independent Arb checker rigorously improve the known lower bound to B > 0.585078819674; the exact value remains open.
PARTIALProved the sharp-order bound for low 2-adic-level and quadratic-energy regimes and exactly certified all 4–6 term spectra through frequency 24; the general \(N^{1/2}\) conjecture remains open, with Be
PARTIALThe live page status is OPEN, but it lists 1 claimed proof and current workers dhyantailor, benk, and Qing_Hong (accessed 2026-07-31).
SKIPPEDProved from scratch the exact radius-one-disk values α(3)=3√3/4, α(4)=1, and α(5)=√(5(5−√5)/32), plus the sharp all-boundary formula β(n)=2 sin²(π/n) sin(2π/n); the asymptotic Erdős problem remains op
PARTIALUnder Elliott’s intended “not all collinear or concyclic” convention, \(f(7)=11\) is proved with an exact integer construction and a from-scratch lower bound; also \(12\leq f(8)\leq18\), with the exac
PARTIALLive page status is OPEN, but “Currently working on this problem” lists Lumantis (accessed 2026-07-29), so no attempt was made.
SKIPPEDLee--Pohoata--Zhu, arXiv:2607.05374v1, source-audited modulo the Hajir--Maire--Ramakrishna/Chebotarev tower input, proves \(P_2(n)<n^{1-c}\) for some \(c>0\) and all sufficiently large \(n\); the full
FOUNDThe problem remains open; dependency-free exhaustive computation and independent arithmetic audits now give the complete exact table \(g_3(N)\) for every \(1\le N\le191\), ending at \(g_3(191)=159\).
PARTIALproved the residue-vector and reflection reductions, exhaustively verified \(A(k)\), \(B(k)\), and greedy \(B\)-optimality for \(1\leq k\leq22\), and derived conditional liminf constants \(1/2\) and \
PARTIALLive page status is OPEN with 0 claimed proofs, but idrissbado is listed under “Currently working on this problem”; the mandatory collision rule therefore requires no attempt.
SKIPPEDLive page status OPEN; it lists 1 full proof claim by Jeff Pickhardt and Omniscience Research Agent (submitted 2026-07-28), so the mandatory no-duplication rule applies.
SKIPPEDThe live page status is OPEN, but it lists one claimed proof—a full proof claimed by Colin Snyder (submitted 2026-07-15)—and “Currently working on this problem” is None, so the task’s mandatory collis
SKIPPEDLive page status OPEN (accessed 2026-07-29), but “Currently working on this problem” lists DaPoWi; per the no-collision rule, no attempt was made.
SKIPPEDThe authoritative live page (accessed 2026-07-29) has status OPEN, 0 claimed proofs, and lists Shadow under “Currently working on this problem”; stopped to avoid colliding with a current worker.
SKIPPEDExplicitly proved the relation when \(\aleph_{\omega+1}\leq2^{\aleph_0}\), repaired and verified the current filter-tail lift, and reduced the remaining hard regime exactly to a ZFC free-cover/tail-fi
PARTIALThe general problem remains open; verified \(C\ge4\) for every plane of order at least 5, exact \(c(\mathrm{PG}(2,3^r))=4\) for all \(r\ge2\), exact checked values through the listed small Desarguesia
PARTIALLive page status OPEN (0 claimed proofs), but “Currently working on this problem” lists Sam_Petkov; stopped to avoid colliding with a current worker.
SKIPPED#1146 remains open; a counterexample must have vanishing simultaneous \(2^m3^n\)-boundary along density-minimizing prefixes, all eventually periodic competitors are excluded, and exact optimization gi
PARTIALProved the critical quadratic-window reduction and excluded every fixed-base digit complement; exhaustively computed the sharp balanced perfect-complement tail distortion for all 2<=m<=30, but the uni
PARTIALLive-page status is OPEN, but “Currently working on this problem” lists apiros3; per the collision rule, no attempt was made.
SKIPPEDproved no fixed-modulus obstruction and exactly certified the stated finite suffixes through \(10^7\) for \(r=3\) and \(10^8\) for \(4\le r\le8\); eventual coverage for every \(r\ge3\) remains open.
PARTIALProved the exact near-simplex band \(f_d(n)\) for \(n\le2d\) (including \(f_d(d+3)=\binom{d+3}{2}-3\) in every dimension), proved \(f_3(7)=15\), and corrected the live page's three-dimensional upper b
PARTIALLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists Tomodovodoo (accessed 2026-07-29); the mandatory collision-avoidance rule requires stopping.
SKIPPEDProved the canonical reduction and a uniform \(O(x/\log^2x)\) upper bound modulo the standard Selberg upper sieve; exactly enumerated 481,051 first-form and 1,021,854 second-form primes through \(10^8
PARTIALExact computational values f(n) are independently reproduced through n=200 (extending OEIS n=45..200), with f(200)=136 and an explicit checked witness; irrationality remains open and requires a unifor
PARTIALProved the exact powerful-core bijection, improved the general located exponent constant from log(3)/3 to log(2)/2 for every odd target, certified f(n)<=5 through 10^14 and f(n)<=1 for odd n through 1
PARTIAL[d] Exactly 646 Carmichael numbers occur up to \(10^9\), verified by two complete kernels and independent audits; [b] the asymptotic problem remains open, with accepted exponent \(0.3389\) and a unifo
PARTIALExact exhaustive computation finds only the five known unitary perfect numbers when every prime-power component is at most \(10^7\); any sixth has a component \(>10^7\), while full finiteness still re
PARTIALproved two uniform closed forms, an exact linear-excess regime modulo Győri's verified packing theorem, a sharp subquadratic estimate, an exact weighted-packing reduction, and the exhaustive table thr
PARTIAL\(h(n)=5\) for every \(34\le n\le40\) via an explicit five-chord family; the cases \(n=38,39,40\) extend the verified published exact table through \(n=37\), while the asymptotic problem remains open.
PROVEDproved the cited f-bounds invert to coefficients 1/2 and 2 (exposing an unsupported factor-two live-page bound), derived 33 <= h_3(6) <= 40 with an explicit fully rechecked upper witness, and reduced
PARTIAL[d] Exhaustive generation plus independently validated colouring certificates give \(f_4(11)=21\), \(f_4(12)=26\), \(f_4(13)=31\), and \(f_4(14)=36\); the general problem remains open.
PARTIALLive page status OPEN, but it lists 2 claimed proofs and current workers mchoi and Woett; per the collision/duplication gate, no attempt was made.
SKIPPEDLIVE PAGE STATUS OPEN (0 formal claimed proofs; “Currently working: None”), but its newest discussion comment (Liam Price, 30 Apr 2026) explicitly says “GPT-5.5 Pro claims a disproof to the first ques
SKIPPEDverified Tenenbaum's exact best general exponent, isolated the zero-density/Type-I–II barrier, recorded the June-2026 monic-cubic advance, and exactly certified \(F_{x^2+1}(n)\ge0.998112594359576\,n^2
PARTIALFor the irreducible cubic \(f(n)=n^3+2\), the \(d\le X\) contribution is rigorously \(2R X\log X\), the problem is reduced exactly to signed endpoint cancellation for \(X<d\le X^{3/2}\), \(R\) has an
PARTIALLive page status is OPEN, but it lists “1 claimed proof for this problem” (a partial proof claimed by KyungMin Han, submitted 2026-07-25 by Dogcake); “Currently working on this problem” is None.
SKIPPEDFGKMT plus \(x=p_k\) rigorously strengthens the cited lower bound to \(H(k)\gg k(\log k)^2\log_3k/\log_2k\); an independent exhaustive checker gives exact \(H(k)\) for \(k\leq14\), but the quadratic u
PARTIALverified and corrected the pre-jump count at the published \(10^{18}\)-range negative extremizer, certified exact decade extrema through \(10^7\), and reduced the RH-conditional \(1/4+\varepsilon\) ta
PARTIALProved, with a uniform exact bound and standalone verification, that every finite direct power \(A_5^r\) satisfies the Herzog–Schönheim conjecture; the general problem remains open.
PARTIALExact carry-factor reduction for all finite P, plus rigorous rational enclosures excluding denominators below 40–51 digits for all three-prime subsets of {2,3,5,7} (and an 18-digit bound for {2,3,5,7}
PARTIALLive page status OPEN, but it lists one full proof claim (Colin Snyder, submitted 2026-07-15, claiming the complete \(c>1\) result with a Lean 4/Mathlib formalisation); currently working and intereste
SKIPPEDproved simultaneous rationality with the sharper endpoint growth \(\liminf a_n^{1/(\sqrt{3/2})^n}>1\), exact target sums, a standalone checker, and a \(\sqrt{5/3}\) counting ceiling for the fixed-wind
PARTIALLive page status OPEN (0 claimed proofs), but “Currently working on this problem” lists daniel_e_ruiz_c and memeister27; mandatory no-collision stop.
SKIPPEDExact radical-support compression proved and a from-scratch rational verifier certifies the sharp bound \(M(n)\leq\sum_{p<n}1/p+1\) for every \(2\leq n\leq1000\); the uniform problem remains blocked b
PARTIAL[D] Every integer \(|n|\le10000\) except \(n=\pm1\) is explicitly ruled out for the always-squarefree variant by a prime-square witness with \(p\le1423\) and \(k\le250270\); within this range existenc
PARTIALExact finite compactness reduction proved; exhaustive checker certifies \(L_N\) through \(N=21\) and normalized minimax \(R_N\) through \(N=12\); the open asymptotic gap remains \(n^2/\log n\) infinit
PARTIALProved the critical-density/lower-energy hierarchy and the exact r=3 block-gluing reduction, and exhaustively certified the minimum cyclic three-basis energy for every modulus 2 through 41; the unreso
PARTIALreduced both questions to exact monotone finite Golomb-ruler profiles, proved the quadratic profile through n=21, and constructed and exactly verified a critical-log-envelope ruler through n=680; no a
PARTIALexact binomial-divisibility reduction proved and q(n,floor log n) exhaustively maximized for every 3<=n<e^21 (unique global maximum q=113), but no uniform constant saving is proved.
PARTIALreduced all uniformities to the exact \(r=3\) core, proved \(f_3(n,7,4)=3,4,6,7,9,12\) for \(7\le n\le12\) with a standalone exhaustive checker, and derived \(f_3(n,7,4)<\frac49n^2+2n\) modulo Gishbol
PARTIALProved the finite-clique hierarchy \(\chi(G)\le\beth_{q-3}(\kappa)\), obtained uniform \(>\kappa\) consequences at \(\beth_\omega(\kappa)\) and its successor-cover extension, reduced exact \(\kappa\)
PARTIALUnder GCH a uniform intersection bound below aleph_omega on any aleph_{omega+1}-sized subfamily forces the desired free set; every counterexample must have kappa full-sized rows, kappa popular points,
PARTIALUnder GCH the first arrow follows from Erdős–Rado, and relative to a huge cardinal GCH is consistent with all symmetric arrows through \(\omega_1^2+1\); the other two GCH arrows and uniformity for eve
PARTIALThe live problem remains open, but k=2 is a verified 1987 ZFC theorem; only k>=3 remains, with an exact triangle-free-cover/block-fusion reduction and a checked sharp C5 block obstruction.
PARTIALReduced #1144 exactly to a weighted squarefree positive-fluctuation lemma plus an \(L^2\)-controlled fractional error, proved the exact variance \((6/\pi^2)N\log N+O(N)\), and exhaustively verified al
PARTIALLive page status is OPEN - $500 with 0 claimed proofs, but “Currently working on this problem” lists VertRule, alansbor, old-bielefelder, and Shang_Yu_Chen (accessed 2026-07-28), so the mandatory no-c
SKIPPEDCertified sharp finite prefix bounds and an ordinary-additive extension of Mangerel's sparse-descent theorem; #1122 remains open at the average-gap and large-prime-spike barriers.
PARTIALThe Izotov candidate is re-proved Sierpiński; any finite cover is certified to require at least 686 primes, including one at least 376843822247957 with period at least 94210955561989, but unbounded le
PARTIALThe live page status is “OPEN (LEAN)”, but it lists “1 claimed proof” and JohanLand as “Currently working on this problem”; per the mandatory collision rule, no mathematical attempt was made.
SKIPPEDProved the self-contained no-pair bounds chi<=7,6,5 for odd girth 5,7,>=9, reducing any d(3,3)<=7 improvement to the C5-layer case; also verified a minimum-order 22-vertex witness for d(3,3)>=6.
PARTIALLive page status is OPEN, but it lists 1 partial claimed proof (Xiyu Hu/hxypqr, submitted 2026-07-23) and names hxypqr as currently working on the problem.
SKIPPEDLive page status OPEN, but comments dated 27–28 April 2026 link a proof and explicitly say that it claims a positive resolution of the second question (“Standard check found no issues”); stopping unde
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists lof310 (accessed 2026-07-28); stopped under the mandatory collision rule.
SKIPPEDProved exactly that $f_d(n)=\binom n2$ for $n\le d+1$, $f_d(d+2)=\binom{d+2}{2}-1$, and $f_d(d+k)=\binom{d+k}{2}-k$ for $3\le k\le d$ (with $f_d(d+3)=\binom{d+3}{2}-3$ also for $d=1,2$); the fixed-d a
PARTIALExact values (modulo named published few-distance classifications) are established for \(f_3(n)\) through \(n=21\) and \(f_4(n)\) through \(n=25\), with an elementary all-\(d\) two-distance interval a
PARTIALproved the conjectured exact value for all n<=15 (modulo the published g(k) classifications), proved D(A)>=ceil(n/3) modulo the planar diameter-graph theorem, and exactly ruled out the standard seven-
PARTIALExact EHS certificates give 120 members through 139 and isolate 140!+1; two independent scans give 21969 Pillai primes through 500000, but both density limits remain open.
PARTIALProved the elementary bound \(A(x)\leq(1+o(1))x/\log\log x\) and exactly certified \(A(200000000)=960\), including 241 values beyond the OEIS \(10^8\) table; the \(x^{o(1)}\) claim remains open at the
PARTIALproved an exact Wilson-reflection reduction and growing endpoint gap, derived a quantitative \((\log Y/\log\log Y)^{1/3}\) lower bound for every cutoff \(c>1/(1+9\log2)\), and independently verified e
PARTIALLIVE PAGE status OPEN with 0 claimed proofs, but “Currently working on this problem” lists jeffhino, so this run stopped before any mathematical attempt to avoid colliding with the current worker.
SKIPPEDLive page status is OPEN, but it lists jif and Vjeko_Kovac as currently working on problem #1054 (accessed 2026-07-28), so the mandatory no-collision rule requires stopping.
SKIPPEDProved that #1053 is exactly equivalent to divergence of the missing-prime sum in (11), classified all squarefree cases, and exhaustively verified the 14 multiply-perfect numbers through \(10^8\); the
PARTIALExact thresholds tau_1=1, tau_2=2, tau_3=6, the bound 9<=tau_4<=13, and a uniform O(2^n/n) deletion guarantee are verified; the fixed-positive-c problem remains open at a quantified cube-specific blow
PARTIALExhaustive computation gives h(3..10) = 6,8,9,10,12,13,14,16 with explicit certificates and an independent labeled cross-check through n=7; the asymptotic constant remains open.
PARTIALLive-page status is FALSIFIABLE; although its tracker says “0 claimed proofs” and “Currently working on this problem: None,” the discussion explicitly reports a claimed complete proof [Mi26] (with a l
SKIPPEDproved a positive-proportion \(L=10^{-10}(\log x)^2/\log\log x\) theorem modulo PPT, reduced all polylogarithmic lengths to GHP-diagonal coverage, and exactly certified \(A(10^9)=691\); the arbitrary-
PARTIALExact radical-kernel bijection proved; all solutions with odd member <= 2^64-1 and omega(odd) <= 3 are exhaustively classified (15 total), but infinitude remains equivalent to constructing infinitely
PARTIALThe live statement is literally false at \(p=2\); for the intended \(p>2\) version, \(G(p)\le211<p\) is independently certified for every prime \(p\le10^7\), two uniform subclasses are proved, and the
FOUNDLive page status on 2026-07-28 is FALSIFIABLE/open, but it lists 1 partial claimed proof (Scott Duke Kominers, submitted 2026-07-25) and current workers Sam_Petkov and skominers, so the mandatory no-d
SKIPPEDExhaustively proved for unordered k=4 representations that M_4(4)=199898912404 (four displayed prime quadruples) and M_4(5)>=5212641500689; the uniform unboundedness question remains open.
PARTIALExactly 75,670 of the first 100,000 values n^4+2 are certified squarefree, the local density is rigorously enclosed in [0.756683702608, 0.756684005282], and the open problem is reduced to the unproved
PARTIALGPY small-gap density plus Stadlmann's mean-square theorem gives \(A(N)\gg_\varepsilon N^{77/100-\varepsilon}\), and two independent exact sieves verify \(A(10^7)=4,212,774\); positive density remains
PARTIALLive page status is OPEN, but “Currently working on this problem” lists KoishiChan (accessed 2026-07-28).
SKIPPEDVia Laishram--Murty's published Grimm-function theorem, \(k(n)<n^{0.46}=n^{1/2-1/25}\) for all sufficiently large \(n\); the main \(\log k(n)\leq(\log n)^{1/2+o(1)}\) conjecture remains open, and the
PARTIALlive gate cleared; proved the quantitative binomial reduction, independently certified exact f(k) for k<=40 modulo the Carmichael-Lehmer cutoff, and constructed f(647134389)>=288; the uniform exponent
PARTIALThe authoritative live page is OPEN with no current worker, but it lists 2 claimed partial proofs (submitted 2026-07-15 and 2026-07-21), so the mandatory no-duplication gate applies.
SKIPPEDLIVE STATUS OPEN, but the 27 Apr 2026 discussion displays a claimed proof that \(h(n)=\Theta(n^{4/3})\), links its proof note, and reports that a standard check found no issues; no current worker is l
SKIPPEDproved an elementary equivalence with uniform size-biased fibre tightness, isolating the exact missing lemma, and exactly certified the sharp finite concentration function through \(x=5{,}000{,}000\);
PARTIALproved the exact regulator/drawdown reduction and certified the sharp bound \(0\le R(x)-x\le46{,}462\) for every integer \(x\le10^8\); the unresolved step is a uniform \(x^{1/4+o(1)}\) drawdown lemma.
PARTIALThe live page is OPEN with 0 claimed proofs but lists jif as “Currently working on this problem” (accessed 2026-07-28), so an attempt would collide with a current worker.
SKIPPEDexact start-anywhere computation gives maximum \(G_2\)-component size 100 and reproduces rooted moat data through \(D=10\); arbitrary bounded step remains open.
PARTIALExact rational verification proves 4.309405275 < m(3) < 4.3094055 and hence a literal finite-x counterexample below 4.3094055; the asymptotic Erdős/Shapiro question remains open.
PARTIALproved the full measurable case (with a perfect-set construction in the null case), repaired the Baire-case diagonal, and isolated the unresolved nonmeasurable/non-Baire regime; the unrestricted probl
PARTIALDual exhaustive computation proves the exact first starts through \(k=13\), the full step function \(F(x)\) for integer \(x\leq20{,}000{,}000\), and no length-14 run there; an elementary reduction iso
PARTIALLive page status OPEN (0 claimed proofs; “Currently working” marker: None), but the 18 March 2026 comment by Przemek Chojecki explicitly reports “on-going research work”; treated as a current worker,
SKIPPEDExact live-normalized computation gives a unique maximum h(n)=12 for n<=10^9 at n=472532614, with 12 factored witnesses and a reproducible verifier; the uniform polylogarithmic upper bound remains ope
PARTIALLive page status is OPEN, but “Currently working on this problem” lists Basile_Beyer_de_Ryke; stopped to avoid colliding with a current worker.
SKIPPED(d) no coprime 4-powerful solution \(a+b=c\) has \(\max(a,b)\leq10^{16}\), a 100-fold extension of the live bound; (a) the \(r\geq6\) infinite construction is independently proved, leaving global \(r=
PARTIALThe live page is OPEN with 0 claimed proofs, but it lists SkyYang as “Currently working on this problem” (accessed 2026-07-28), so the mandatory no-collision rule requires stopping without a mathemati
SKIPPEDExact certificate checks give only n={1} for 2^n-1 and n={3} for 2^n+1 through n=1000, and only n={4,5,7} for n!+1 and n={2} for n!-1 through n=139; LTE reduces each exponential branch to a uniform no
PARTIALelementary density-one bounds and exact \(n\le10^9\) tables for \(\ell=2,3\) are verified; the uniform first and third questions remain open at the explicit radical/simultaneous-powerful finiteness wa
PARTIALverified the July-2026 disproofs from explicit Odd/Witt constructions, certified h_3(4)>=71 and h_3(120)>=1,730,521, proved h_t(2)=2t+2, and isolated the remaining 253/225-versus-3/2 t=3 gap.
PARTIALLive page status OPEN; it lists current workers johnseibert19, fschumann, and aminb_el (with 0 claimed proofs, though one comment links a purported negative-answer preprint), so the mandatory no-colli
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists lof310 (accessed 2026-07-28), so the mandated collision-avoidance rule forbids an attempt.
SKIPPEDThe live page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists Svyable and satsun93 (accessed 2026-07-28); per the collision-avoidance gate, no mathematical attempt
SKIPPEDreduced #929 exactly to the primorial Jacobsthal bound \(Y(x)\le x^{1+o(1)}\), extracted the sharper proved bracket \(\sqrt{k}\ll S(k)\ll k\log_2k/(\log k\log_3k)\), and independently certified exact
PARTIALTao--Teräväinen gives the predicted ordinary density outside a zero-log-density set of scales; the supplied independently checked census is exact through 10^7, and the remaining fixed-shift all-scales
PARTIALproved an elementary O(sqrt(X)) candidate bound, classified the max-exponent-2 and two-prime slices, ruled out the standard infinite powerful recurrence beyond n=8, and independently verified exactly
PARTIALProved the uniform bound |h(n+1)-h(n)|<=1, gave an exact multiplicity/short-interval reduction and a three-way exact table through 10^9; the main asymptotic remains open at the stated harmonic-bin Poi
PARTIALVerified \(48\leq f(4)\leq67\); \(P_{67}\) is an explicit \(S_4\) tournament with a standalone exhaustive checker, while the asymptotic factor-\(n\) gap remains open.
PARTIALVerified \(m_{11}(5)=m_{12}(5)=66\), certified an edge-critical 51-edge 3-chromatic construction, and reduced the current \(32\le m(5)\le51\) gap to explicit finite set-cover windows; the asymptotic p
PARTIALproved the uniform bound f(2n)/f(n)>2 and a Bang–Zsigmondy weighted reduction, with an independent exact certificate through n=138; full divergence remains open.
PARTIALProved the necessary condition \(\sum_i1/n_i<\infty\) for the final scale question and an iff theorem for \(b_n\asymp n\log_2n\log_3n\,L(\log_2n)\), with an explicit independently checked primitive co
PARTIALExact smooth-centre reduction; all k=2 starts through 10^30 and k=3 starts through 10^12 classified, with last bad starts 4372 and 8615 respectively, but no uniform all-large-n lemma.
PARTIALexact interval reduction and exhaustive verification give \(v_0(n)\ge3\) for every \(14{,}433{,}527\le n\le50{,}000{,}000\), with exactly 2,112 smaller-prefix exceptions and last \(14{,}433{,}526\); e
PARTIALCorrected the live comment's off-by-one error, proved the exact baseline-plus-hypermatching reduction, and computationally verified \(G(n)=G_{\le2}(n)\) for every \(n\le5000\) plus exact samples throu
PARTIALLive page status OPEN; 0 claimed proofs; currently working on this problem: KStar (live page accessed 2026-07-28).
SKIPPEDThe live page is marked OPEN but lists 1 partial claimed proof (Liam Price using GPT-5.6 Sol Pro, submitted 2026-07-18) asserting that every infinite sum-free sequence satisfies \(\limsup_{n\to\infty}
SKIPPEDLive page status is OPEN (0 formal claimed proofs; no current worker), but its discussion contains DavidTurturean’s explicit 24 April 2026 claim of a total refutation for every k >= 3, updated 2 May 2
SKIPPEDproved the exact finite-union/LCM reduction, derived the rigorous lower bound \(\liminf d_t\log t\ge ce^{-\gamma}\) from Weingartner's theorem (forcing any asymptotic exponent \(c_2\le1\)), and indepe
PARTIALThe live page status is OPEN, but it lists sproutseeds as “Currently working on this problem”; the task forbids colliding with a current worker.
SKIPPEDThe live page status is DECIDABLE and states that Sawhney solved the problem for all sufficiently large \(N\); it lists 0 claimed proofs and no current worker.
SKIPPEDexact standard-library search certifies the finite extremal table \(f(N)\) for every \(1\leq N\leq70\), audits the elementary \(2/3\) bound and Freud construction, and isolates uncontrolled overlap am
PARTIALProved an exact divisor and Eisenstein-norm reduction, proved the conjectured polylog bound uniformly when the number of prime divisors p=1 (mod 3) is bounded (with a complete sharp classification whe
PARTIALclassified all possible bounded-\(\omega\) infinite families, reproduced the exact \(a=1\) list through seven prime factors, and isolated the unresolved unbounded-prime-support product-divisibility le
PARTIALexactly ten sharp-constant solutions \(n\leq10^9\), with a rigorous all-\(k\) checker and \((q,2q-1)\) prime-pair obstruction; #826 remains open at the near-critical \(\log\tau\) concentration lemma.
PARTIALLive page status is OPEN, but “Currently working on this problem” lists SamKorsky (accessed 2026-07-28); stopping to avoid colliding with a current worker.
SKIPPEDexact computer-assisted table \(l(1..10)=1,2,2,3,3,3,4,4,4,4\) and \(4\leq l(n)\leq5\) for \(11\leq n\leq14\); both asymptotic questions remain open.
PARTIALexact page-formulation values are h(1..19)=1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4, with 4<=h(n)<=5 for 20<=n<=23; the asymptotic exponent gap remains open.
PARTIALThe live page status is OPEN, but it lists “1 claimed proof for this problem” (and “Currently working on this problem: None”), triggering the mandatory no-duplication stop rule.
SKIPPEDBob wins the biased \(1{:}2\) clique game for every \(4\le n\le7\); \(n=4,5\) have elementary strategies and \(n=6,7\) have exact stronger triangle-avoidance certificates, but no uniform \(n>3\) argum
PARTIALLive page status is OPEN, but it lists 2 claimed proofs and a current worker (mthiim).
SKIPPEDexact 15-point obstruction/table and base-family classification verified; #774 remains equivalent to the unresolved uniform finite bound \(C(\delta)<\infty\).
PARTIALLive page status OPEN (0 claimed proofs), but “Currently working on this problem” lists Aron and ogroth; stopped to avoid colliding with current workers.
SKIPPEDLive page status is FALSIFIABLE/open (0 site-registered claimed proofs), but it lists jif as “Currently working on this problem” and its 28 Feb 2026 discussion also records arXiv:2410.13840 as a claim
SKIPPEDExact reproducible checks find no counterexample through n=14 and reduce the first open order n=25 to five SAT instances (maximum degree 13 through 17); the uniform conjecture remains open.
PARTIALLive page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists ryin and boolean_matrix (accessed 2026-07-28); stopped before mathematics to avoid colliding with current wor
SKIPPEDRigorous modulo MRSTT Proposition 1.13, \(1/6\leq\liminf S(n)/\log\log n\leq\limsup S(n)/\log\log n\leq5/6\); exhaustive rational enclosures verified every \(100\leq n\leq10^7\), but the uniform recip
PARTIALverified Godsil--McKay through \(k=o(n^{6/7})\), proved a uniform permanent reduction/logarithmic bound, and independently computed exact reduced counts through \(n=6\) plus \(R(4,7)=1{,}293{,}216\);
PARTIALProved exactly that ex(n;K_{r,r}) equals C(n,2) for n<2r, C(2r,2)-r at n=2r, and C(2r+1,2)-(r+2) at n=2r+1, with exhaustive independent checks through r=4; the asymptotic question remains open.
PARTIALThe authoritative live statement is literally false for \(G=K_2\) since \(\operatorname{ex}(n,K_2)=0\); for the intended nondegenerate problem, the report proves an explicit infinite \(K_{2,t}\)-core
FOUNDProved the exact covering-design reduction, closed three near-diagonal finite regimes (including all ex_3(n,K_{n-2}^3)), and independently certified ex_3(n,K_4^3)=3,7,14,23,36,54 for n=4,...,9; no com
PARTIALThe live page (accessed 2026-07-28) has status FALSIFIABLE/open, lists 2 claimed partial proofs (one accepted by the site), and names conglu, Woett, and ster as currently working; the mandatory no-dup
SKIPPEDExact multiplier-cost reduction and independently verified sharp terminal-prime bounds for every prime-chain length 1 through 14; the uniform exceptional-path lemma needed for either asymptotic questi
PARTIALFor the strict live-page problem, \(G_2(n)\ll n^{2/3}\) follows from standard Erdős–Turán/van der Corput bounds, the exact remaining one-sided discrepancy condition is isolated, and exhaustive indepen
PARTIALIndependently certified \(Y(x)=h(\pi(x))-1\) and the exact values through \(x<43\) (in particular \(Y=73\) on \(41\le x<43\)); a fixed positive interval-sieve survivor bound (5.3) would prove \(Y=o(x^
PARTIALFor epsilon=1/10 and cutoff K=16, exactly 94 admissible n occur through Q_101+15 and all are at most 225, while the infinite problem reduces to an every-interval multiscale gap lemma not supplied by k
PARTIALLive page status OPEN, but it lists “1 claimed proof for this problem” (with “Currently working on this problem: None”), so the mandatory claimed-proof stop rule applies.
SKIPPEDElementary proof gives continuum many incongruent maximizers at n=6; modulo AMP25's exact u(n) theorem the same holds at n=8,9,12,21, while no uniform extremality-propagation lemma is known.
PARTIALExact \(f(n)\) for every \(3\le n\le21\), plus a verified \(C=2\) construction for every \(q^2\le n\le q^2+q+1\) with prime-power \(q\); the uniform problem remains blocked by gaps between projective-
PARTIALexact CRT reduction plus a complete k=2 enumeration proves the sharp cutoff q(n,2)<=199 for every 1<=n<68979941211663467981891685180356798343731381481900118460455756739438, with q(N,2)=211 at equality
PARTIALLive page status is OPEN, but it lists 1 claimed full proof (submitted by Colin Snyder on 2026-07-15); per the mandatory collision rule, no mathematical attempt was made.
SKIPPEDThe disjoint diagonal half-shift has an exact odd-square-sum distance set and is the unique finite-family minimizer in 848 arithmetic candidates for every \(4\le m\le64\), but Landau–Ramanujan rigorou
PARTIALPublished few-distance classifications plus exact witnesses determine \(f_{\rm cvx}(n)\) for every \(4\le n\le20\) (with a jump from 3 to 5), and an elementary regular-pyramid family attains \(\lfloor
PARTIALExact elementary/algebraic certificates prove \(\phi(n,3,3)=3,3,5,6\) for \(n=3,4,5,6\); the planar asymptotic question remains open.
PARTIALThe literal live-page formulation is false and its exact sharp minimum is \(D(X)=\lfloor n/2\rfloor\), attained for every \(n\) by the regular \(n\)-gon; historically intended no-four-concyclic pinned
PROVEDProved \(f(2)=1,\ f(3)=1,\ f(4)=2,\ f(5)=f(6)=3\), established \(3\leq f(7)\leq4\) and \(4\leq f(8)\leq5\), and supplied exact no-four-concyclic constructions and checks; the asymptotic \((1/3+c)n\) q
PARTIALProved \(g(n)\le n-2\) for every \(n\ge7\) and, by exact nested \(\mathbb Q(\sqrt3)\) constructions, \(g(7)=5\), \(g(8)=6\), and \(g(9)=7\); the asymptotic question remains open.
PARTIALMandatory non-collision stop: the live page status is open (“VERIFIABLE - $44”) with 0 claimed proofs, but “Currently working on this problem” lists Ritvik_Nayak, pommeret, ScottHughes, and will0708 (
SKIPPEDThe explicit 12-uniform parity family on 22 vertices has computationally certified property (7,2) and elementary transversal number 10, proving 10 <= f(12,7) <= 11; both asymptotic questions remain op
PARTIALproved the exact boundary formula for every t at n<=2t and independently certified g_3(7)=17, g_3(8)=23 (live thresholds f(7;3)=18, f(8;3)=24); the asymptotic problem remains open.
PARTIALRigorous modulo the published DMMS lemmas, the final parameter argument improves the known upper bound to f(n)=O(n(log n)^7); the independent checker passes the construction, f(n) through 7, all const
PARTIALexact M_2(N)=ceil(N/2)+ceil(floor(log_2 N)/2) is certified for every N<=256, M_t(N) is proved exactly when 3t>N, and a precise Hall-expansion lemma isolates the unproved uniform t=2 step.
PARTIALFor the unrestricted live statement, proved the threshold reduction \(p(h)/2^h\to0\), supplied directly checked witnesses, and computationally established \(H(n)=0,1,2,3,3,3,4,4,4,4,4\) for \(1\le n\l
PARTIAL(b) Morris--Sahasrabudhe--Verstraëte, arXiv:2607.16118v1, gives the uniform answer \(f(n)=\Theta(\sqrt{n\log n})\); independently (d), exhaustive checks give \(f(1),\ldots,f(10)=1,2,2,3,4,4,4,5,5,6\).
PROVEDproved the exact table t(3..7)=(1,1,1,2,2) and gave a checked explicit family with tau=s+1 for every r>=5s+1+floor((s-1)/3); the general problem remains open.
PARTIALproved the sharp \(K_5\)-free odd-clique bound for \(d=5,10,15,20\), with uniform-in-diameter finite certificates and a standalone verifier; the general problem remains open.
PARTIALConstructed and exactly checked a 10-point proper locally four-distance set, proving \(f(10)=4\) modulo the published sharp theorem \(L(3)=8\); the asymptotic Erdős problem remains open.
PARTIALLIVE PAGE status OPEN (accessed 2026-07-28); it lists “1 claimed proof for this problem”—a partial proof claimed by Lezhe Gao, submitted 2026-07-22 with external proof at Zenodo record 21475089—so Ste
SKIPPEDEvery rayless instance reduces rigorously to a full-type finite-kernel column graph with a ccc finite-independent-set poset; the remaining CH-frontier obstruction is the full-order kernel avoidance pl
PARTIALexact e(n,r) is verified through n=10 and closed forms are proved for r=n-2 and r=n-3, but both fixed-r asymptotic questions remain open.
PARTIALRelative independence is verified modulo Garti--Hayut and Jech; additionally, any first bad cardinal is proved regular and \(\aleph_0\)-closed, sharpening the linked threshold note, while the exact co
PARTIALThe live page is OPEN - $250 with 0 claimed proofs, but lists Sam_Petkov as “Currently working on this problem.”
SKIPPEDReduced every unresolved positive instance to uniform three-way amalgamation of Schipperus game strategies, and proved that every endpoint-compatible alternating block pattern triangle-amalgamates (wi
PARTIALExact computational values F(1..10)=0,1,3,6,9,12,16,19,23,27 are certified with explicit lower graphs and 918,981 exhaustive upper checks; the asymptotic gap remains open.
PARTIALThe literal live H2 assertion is false, rigorously modulo LUW Proposition 2.1: explicit D(5,q) graphs have girth at least 10 but require Omega(q^2) H2 edges; this does not settle the sparsity-qualifie
PROVEDExact exhaustive path-number distributions and the stronger odd-semi-clique classification are verified for every connected simple graph on at most 8 vertices; no uniform proof or counterexample is ob
PARTIALThe live page is still open (display badge “DECIDABLE”) and lists no current worker, but it lists 1 partial claimed proof—Rafik Zeraoulia’s claimed verification through \(n\le 19\), with a visible com
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists SkyYang (accessed 2026-07-28), so the mandatory no-collision rule requires stopping.
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists Dharun; stopped to avoid colliding with a current worker.
SKIPPEDThe live #575 statement is uniformly false for \(\mathcal F=\{K_{1,2},2K_2\}\): \(\operatorname{ex}(n;\mathcal F)=1\) for \(n\ge2\), while both member extremal numbers grow linearly.
FOUNDindependently proved and reverified the exact table \(f(n)=0,1,2,3,5,6,8,10,12,15,16,18,21,23,26\) for \(1\leq n\leq15\); the asymptotic factor-\(\sqrt2\) upper-bound gap remains.
PARTIALWoodall gives the exact all-\(k\) range \(n<2k\) and \(2k\le n\le4k-3\), and an independent exhaustive check verifies the first three \(C_8\) cases, but the required asymptotic construction remains op
PARTIALcombining the 2026 JLY theorem with KKL densification proves the family (5), including \(29/18\), and certifies all 46 reduced rationals of denominator at most 12; the first gap in the audited suite i
PARTIALExact same-order extremal numbers are proved for every tree (and independently checked for all 106 ten-vertex trees); page \(k=9\) is reduced to 41 tree types and critical hosts with \(n\ge15\), \(e=4
PARTIALproved h(1)=1, h(2)=2, h(3)=3, h(4)=4; gave and independently checked an exact family with h((m+1)^d-2) <= (m+1)^d-(d+1), including h(7)<=6, plus the exact 3x3-grid table; the unrestricted asymptotic
PARTIALmodulo Bertrand's postulate and Mertens' prime-reciprocal theorem, \(M_r(N)\asymp\log N\min\{1,r/\log\log N\}\) uniformly for \(r\ge2\); exact finite bounds, a cap-two construction, a cold Lean audit,
PROVEDStatus OPEN (“This is open, and cannot be resolved with a finite computation”); the mandatory page check found current workers SharkyKesa and SamKorsky in the latest retrievable indexed problem-page r
SKIPPEDThe live page is OPEN with 0 formal claimed proofs and no current worker, but its 2026-07-28 discussion comment explicitly reports that GPT-5.6 Pro claims a negative resolution (with a linked manuscri
SKIPPEDVerified \(\chi(\mathbb F_{11}^2)=5\) and, modulo Hensel/DRAT checking, \(\chi(\mathbb Q(\sqrt3,\sqrt5,\sqrt{11})^2)=5\); hence no 6-chromatic unit-distance witness can lie in that field, while \(5\le
PARTIALVerified \(f(22)=276\), an explicit \(278\)-point construction in \(\mathbb R^{23}\), the exact \(d\leq8\) table, and the revised reduction \(f(d)=\max\{g(d),s(d)+1,s(d-1)+3\}\); the first remaining c
PARTIALA null-initial well-order gives an explicit counterexample, so the first question is negative under add(N)=cov(N) or non(N)=c; the live joint problem remains OPEN.
PARTIALindependently proved and checked the exact table through n=9 and the sharp 5/9 bound for every three-colour-multiset blow-up; the unrestricted density gap remains, and the live page's 0.5611666 decima
PARTIALproved the rational-dependence subcase constructively, reduced the hard pair \((\sqrt2,\sqrt3)\) to exact Pell residuals, certified its complete record table through \(10^7\), and constructed a checke
PARTIALProved an exact classification for all solutions with at most one odd prime divisor, verified explicit extension certificates, and exhaustively enumerated \(n\leq20{,}000{,}000\) for \(|k|\leq20\); th
PARTIALLive page status OPEN; it lists a partial proof claimed by Xiyu Hu (submitted by hxypqr on 2026-07-23) and names hxypqr as currently working on the problem, so the mandatory no-duplication rule applie
SKIPPEDThe seed \(3,5\) is exactly certified through \(10{,}000{,}000\) terms with \(q_{10^7}=437{,}662{,}243\) and sharp gap bound \(3270\) on that range; infinitude remains open because the required adapti
PARTIALLive page status is OPEN - $10 with 0 claimed proofs, but “Currently working on this problem” lists Tomodovodoo; stopping to avoid colliding with the current worker.
SKIPPEDproved the elementary uniform mean bound \(\sum_{n\le x}|D'_n|\le(2\pi/\sqrt3)x\) and independently certified all values through \(10^6\), finding new records 25, 29, and 32 at \(240240,277200,942480\
PARTIALExact anchor/knapsack certificates rule out every x<=7012 under 1<=n<x; 7013 is only the first relaxation admission, and the asymptotic two-colour problem remains open.
PARTIALThe authoritative live page is OPEN with 0 claimed proofs, but lists Svyable as “Currently working on this problem”; per the collision rule, no mathematical attempt was made.
SKIPPEDLive page status is OPEN (0 formally listed claimed proofs; currently working: None), but the live discussion has a 4 May 2026 comment by DavidTurturean explicitly claiming an unconditional solution o
SKIPPED\(\liminf q_n/n^2\ge1/(S-1)>0.543448148339064\), and exactly \(M(100000)=250\); the required limit remains open and reduces to linked-block sparsity (7).
PROVEDexact CRT reduction and density verified; n_k certified for 1<=k<=110; n_184>10^14 certified; the unresolved step is a uniform initial-interval hit bound such as n_k<=D_k^{-C}.
PARTIALLive page status is OPEN, but it lists 1 claimed proof—a full proof claimed by Colin Snyder (submitted 2026-07-15 01:30:36)—so the mandatory collision rule forbids an attempt.
SKIPPEDThe Mills-realizable coloring \(c(n)=\sum_{q\notin\{2,5,7\}}v_q(n)\bmod5\) has first zero triple \(37,329,226\), proving \(\Lambda(5,3)\ge37,329,226\) (and the same for odd \(5\mid k\)); finiteness re
PARTIALexactly 100 exceptions are certified for 5 <= n <= 10^10 (last 267680), with an elementary reduction and standalone checker; the uniform semiprime-gap step remains open.
PARTIALLive page status is OPEN, but it lists 2 claimed proofs (a full proof claimed by Samuel Korsky on 2026-07-20 and a partial proof claimed by him on 2026-07-18); “Currently working on this problem” and
SKIPPEDProved an elementary canonical descent forest and density-neutral recurrence for all holes; independently verified exactly 4,302 holes and all gap runs through \(2^{25}\), but the uniform multiplicity
PARTIALLive-page status is OPEN with 0 claimed proofs, but Rafikzeraoulia2025 is listed as “Currently working on this problem”; stopped before mathematical work to avoid colliding with a current worker.
SKIPPEDProved an elementary uniform prime-cofactor product formula and exact necessary-and-sufficient reductions (11)–(12), and exhaustively verified all 999,998 values of F(log n,n) through 10^6; the missin
PARTIALRigorous modulo Ford (1998/2013), the ratio has limsup strictly above 1, so any existing limit is >1; existence remains open, with exact certified data through \(x=10^7\).
PARTIALexact primitive-totient reduction and a Ford-modulo dyadic block-average limit proved; two independent sieves compute \(V\) and primitive counts through \(2^{25}\), while the one-step error-increment
PARTIALLive page status is OPEN, but its 13 July 2026 discussion comment explicitly says “Here’s the full solution” and links erdos415-sol.pdf; stopping under the no-duplication rule despite the page’s forma
SKIPPEDExact frontier computation proves coalescence for every pair of starts at most 10^10; the open uniform step is to force single-component square frontiers for arbitrarily large k.
PARTIALLive page status OPEN (last edited 17 April 2026; accessed 2026-07-28), with 0 claimed proofs, but “Currently working on this problem” lists conglu and RaziqStark; stopped to avoid colliding with curr
SKIPPEDIndependently proved computationally that the sigma-orbits of 2 and 5 have no common value at most 10^342, extending their published 10^200 separation; no uniform invariant was found, and the next fre
PARTIALThe literal live statement has explicit infinite transient families outside the page's conjectured odd-part list, including every \(2^a3^b\) with \(a,b\geq1\); the exact basin/cycle reduction and a \(
FOUNDLive page status OPEN (accessed 2026-07-28), with 0 claimed proofs and RomanLeLan listed under “Currently working on this problem”; stopped to avoid colliding with the current worker.
SKIPPEDLive page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists Svyable; stopped to avoid colliding with the current worker.
SKIPPEDThe live Erdős Problems #406 page (accessed 2026-07-28) has status OPEN, 0 claimed proofs, and lists `chamb` as “Currently working on this problem” (also “Interested in collaborating”), so the mandato
SKIPPEDLive page status is OPEN, but Sky-Yang is listed as currently working on Erdős problem #404 (accessed 2026-07-28), so the mandatory no-collision rule applies.
SKIPPEDLive page status OPEN; it lists SamKorsky and arkyang as currently working on the problem, so the required no-collision rule applies.
SKIPPEDThe live page displays the problem as FALSIFIABLE/open, with 0 site-filed claimed proofs and no current worker, but its comments list Ahmad Sabihi's paper “A short solution for Brocard-Ramanujans’ pro
SKIPPEDLive page status is OPEN, but “Currently working on this problem” lists jdehorty and SharkyKesa (checked 2026-07-28 via the required Bright Data browser path).
SKIPPEDExact exhaustive computation extends A388302 from n=79 through n=100 (in particular f(80)=76), with a complete geometric-mean reduction and standalone verifier; the uniform asymptotic problem remains
PARTIALLive page status is OPEN, but it lists one full claimed proof (Shouqiao Wang, submitted 2026-07-19), so the mandated no-duplication rule applies.
SKIPPEDProved an elementary prime-gap reduction and, modulo Granville–Ramaré plus PNT, isolated all sufficiently large solutions to a thin prime-free edge regime; exact certified searches found only the nine
PARTIALExactly 100 values n<=10^9 have F(n)<=n, all equalities and the last n=267680; also F(n)-n>=16384 for 521111630<=n<=10^9, while the uniform rough-semiprime lemma needed for either asymptotic question
PARTIALLive page status is OPEN with 1 claimed proof—a partial proof claimed by Rafik Zeraoulia and submitted 2026-07-25—while “Currently working on this problem” is None (accessed 2026-07-28); the mandatory
SKIPPEDExplicit bad intervals are rigorously certified through width 13; exact first starts are computed for widths 0–8, and an audited exhaustive search gives \(999999992\leq U_9\leq4928180396\), but neithe
PARTIALExactly certified max_{n<=10^8} f(n)=f(3250)=1.1792429057944813969965..., found the omitted Sander log-weighted theorem, and reduced the unresolved issue precisely to a uniform small-prime/digit-depth
PARTIALexact exhaustive enumeration gives 14,273 positive solutions through \(10^{100}\), extending the public \(10^{70}\) table by 12,900 verified values, but supplies no uniform infinitude argument.
PARTIALRigorous modulo Laishram--Shorey (2006) plus a twice-reproduced exact computation, Grimm's conjecture holds uniformly for every block length k <= 59; the full problem remains open.
PROVEDLive page status is OPEN with “1 claimed proof for this problem” (a partial proof claimed by Rafik Zeraoulia, submitted 2026-07-27 03:24:16); “Currently working on this problem” is None, but the manda
SKIPPEDThe problem remains open; primary literature gives the omitted unconditional Hickerson classification through \(n\le e^{80}\), while an exact valuation recursion independently finds only the four know
PARTIALProved an exact monotone-triple reduction, exhaustively verified all \(n\leq10^8\) with sharp tail discrepancy bounds, and isolated all-scales fixed-shift exchangeability as the missing lemma; the den
PARTIALModulo Luca–Najman’s corrected completeness theorem, the exact maxima and counts for every prime smoothness bound through 97 are verified, with last \(F(n)<100\) at \(n=9,591,468,737,351,909,375\); th
PARTIALThe live page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists sproutseeds (accessed 2026-07-28), so the mandatory non-collision rule requires stopping.
SKIPPEDLive page status is VERIFIABLE and lists “1 claimed proof for this problem” (currently working: None), so the mandatory no-duplication rule applies.
SKIPPED\(S(10^{16})=26\) is exhaustively certified (five pairs have neither endpoint square), the best located theorem is \(S(x)\ll_\epsilon x^{29/100+\epsilon}\), and the unresolved step is a uniform polylo
PARTIALExact from-scratch enumeration proves there is no square-centered powerful triple \(x^2-1,x^2,x^2+1\) for \(x\le10^{17}\) (center \(\le10^{34}\)); the unrestricted problem remains open because no unif
PARTIALThe live page is OPEN but lists 1 claimed full proof (Principia Math, submitted 2026-07-23) and current workers ruiliangli and Basile_Beyer_de_Ryke, so no attempt was made.
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists AnimishSharma; stopped to avoid colliding with a current worker.
SKIPPEDexact computer-assisted thresholds are proved for a 16-cell rectangle, including greatest exception 558 for the irrational pair (10√2,10√3), and the remaining uniform finite-interval lemma is isolated
PARTIALLive page status OPEN; SSD117 and Sprite144 are listed as currently working on problem #348, so no attempt was made.
SKIPPEDLive page status OPEN; it lists 1 claimed proof (a full proof/counterexample claimed by Liam Price, submitted 2026-07-15) and a current worker (SkyYang), so the mandatory no-duplication/no-collision r
SKIPPEDauthoritative live-page status is OPEN, but “Currently working on this problem” lists old-bielefelder (0 claimed proofs); stopped to avoid colliding with a current worker.
SKIPPEDproved an elementary all-future certificate, certified the exact 1176/224 square-seed tail and 4094 of 4095 seeds in [12], with one precisely bounded unresolved seed.
PARTIALproved a deletion-robust order-3 basis of restricted order 6, completely resolved both finite-deletion questions for eventually periodic bases, and isolated the nonuniform deletion-threshold lemma sti
PARTIALLive-page status OPEN (accessed 2026-07-28), with no current worker, but the page lists 1 claimed proof—a full proof claimed by Colin Snyder (coffeewithcolin), submitted 2026-07-15, asserting \(\lim_{
SKIPPEDExact computation proves G(73)=G(79)=131,486,759 and F(131,486,759)=83; a congruence construction gives F(p)≫log p infinitely often modulo Linnik, while the uniform n^{o(1)} question remains blocked b
PARTIALProved an explicit zero-upper-Banach-density realization \(D(A)=S\) for every \(S\subseteq\mathbb N\), independently verified positive-upper-Banach sufficiency and the packing reduction, and ran the s
PARTIALLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists AnimishSharma (accessed 2026-07-28); per the no-collision rule, no attempt was made.
SKIPPEDLive Erdős Problems page status is OPEN with 0 claimed proofs, but it lists “Currently working on this problem: Aron” (accessed 2026-07-28), so the mandatory collision rule requires no attempt.
SKIPPEDPublished literature plus a checked deduction proves the full asymptotic for every k>=26, improves the lower exponents for k=23,24,25, and gives an exact N=200 table; the verifiable open range is k=3,
PARTIALVerified \(f_{3,3}(10^9)=100735175\) by two independent exact algorithms, derived \(f_{k,3}(x)\sim \Gamma(1+1/k)^3x^{3/k}/(6\Gamma(1+3/k))\) for every \(k\ge26\) from Salberger, and isolated the suffi
PARTIALLive page status is OPEN, but “Currently working on this problem” lists boolean_matrix; the mandatory collision rule therefore forbids an attempt.
SKIPPEDProved an NP-complete upper-bound-attainment subproblem, derived exact common-core and divisibility-chain formulas, and exhaustively certified all 511 nontrivial sets with maximum modulus at most 10;
PARTIALthe Ismailescu--Son candidate is re-certified, and any covering integer must contain at least 59 distinct primes above 10^7 (and outside its 30 covering primes), but ruling out every finite cover rema
PARTIALLive page status OPEN (accessed 2026-07-28); it lists “1 claimed proof for this problem,” specifically a partial proof claimed by Rafik Zeraoulia using OpenAI GPT-5.6 Thinking and submitted 2026-07-27
SKIPPEDLive page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists arkyang (accessed 2026-07-28); stopped to avoid colliding with the current worker.
SKIPPEDproved a sharp explicit formula for every fixed index in the large-seed regime and an O(log n) exact first-obstruction reduction; independently certified A(4) through a_20000, corrected two indexing/p
PARTIALProved fixed-bound factorial-tail separation and Hausdorff dimension zero, reduced a positive factorial answer to open problem #68, and exactly certified that every rational sum for all admissible $\{
PARTIALLive page status is OPEN with 0 formal proof claims and no current worker, but its 9 May 2026 comments explicitly report a checked, Lean-formalized disproof of the second question; the claimed/falsifi
SKIPPEDexact greedy constructions now verify every 1<=n<=100000, correct TUZ20's n=5588 table row, and reduce the uniform problem to a specific modular orbit; the all-n and rational-continuum questions remai
PARTIALLive page status OPEN; it lists 1 claimed proof and Hanziwww as “Currently working on this problem” (accessed 2026-07-28), so the mandatory non-duplication stop applies.
SKIPPEDLive page status OPEN; it lists 2 claimed proofs and Steve_Fan as currently working, so the mandatory no-duplication/no-collision rule applies.
SKIPPEDFor the smallest open case k=5, an exact six-term reduction and independently certified 51-digit computation prove that any rational value has reduced denominator greater than 100000000000000000000000
PARTIALThe main irrationality question remains open; exact reduction (4) isolates eventual periodicity of weighted prime-gap tails, and a reproducible Farey/continued-fraction certificate proves that any rat
PARTIALThe live page status is OPEN, but “Currently working on this problem” lists Terrez1000 and VPendyala (checked 2026-07-28), so the mandatory no-collision gate applies.
SKIPPEDLive page status OPEN; “Currently working on this problem” lists ekalvi (checked 2026-07-28), so the mandatory no-collision rule forbids an attempt.
SKIPPEDLive page status is OPEN with 0 claimed proofs, but “Currently working on this problem” lists Svyable; stopped to avoid colliding with a current worker.
SKIPPEDLive page status “FALSIFIABLE - $250”; “Currently working on this problem” lists KMendoza, dahlkebj, and Sam_Petkov (page accessed 2026-07-28 UTC), so the mandatory no-collision rule applies and no ma
SKIPPEDExplicitly verified a 32-point integer \(n=7\) lower-bound witness, certified the sharp \((7,4,7)\) constrained regime in a stronger signotope relaxation with two SAT engines, and reduced the open 33-
PARTIALThe live page is open with 0 claimed proofs, but its “Currently working on this problem” marker lists JineonBaek and Vugar_Guliyev; per the mandatory no-collision rule, no mathematical attempt was mad
SKIPPEDLive page status is “OPEN - $100”, with 0 claimed proofs, but “Currently working on this problem” lists rubicon (accessed 2026-07-28); the mandatory collision rule therefore forbids an attempt.
SKIPPEDproved an elementary primorial-strengthened bounded-strip theorem and logarithmic record-drift bound, derived long finite monotone staircases from a named almost-prime theorem, and supplied an exact s
PARTIALLive-page status is OPEN with 0 formal proof claims and no current worker, but the 30 April 2026 discussion explicitly claims a Matomäki–Radziwiłł proof, links a solution note, and includes a later “s
SKIPPEDExact standard-library certification of the direct invariant \(\mu(N)\) for every \(1\le N\le40\) (maximum \(1.422032936328\ldots\)); the uniform arbitrary-shift large-prime lemma needed for Erdős #12
PARTIALarXiv:2607.17333v2 gives an exact affirmative solution to #1199; the current proof audit found no gap modulo its named Stone–Čech facts and Hindman's 1979 Corollary 2.10, and an independent exhaustive
FOUNDYounis's 2024 all-interval theorem plus the classical Dickman theorem proves #1184 for every fixed \(1<\alpha<30/17\) unconditionally (and for \(1<\alpha<2\) under RH); the endpoints and larger \(\alp
PARTIALVerified F(7)=11 and f(7)=16 by a fresh exhaustive checker, recovered the published table through n=12, and sharpened the live-page existential bound to n^(3/2)sqrt(log n) << f(n) << n^(3/2)log n via
PARTIALExact labelled distributions through n=7 (plus the n=8 chi<=3 prefix), two all-n tail formulas, an eventual-unimodality reduction for question 1, and an exact fixed-window reformulation for question 2
PARTIALReconstructed the known sharp worst-prime-tuple value \(\lceil2\sqrt u\rceil\) for \(2<\alpha<3\), certified explicit \(\alpha=5/2\) examples for \(u\leq16\), proved current arithmetic-Kakeya lower tr
PARTIALExact modular covering proves there is no solution for \(2^{128}\le n\le340282371933923199876807076826562457784\); the uniform Erdős problem remains open.
PARTIALproved the exact equivalence \(R_N\to0\iff\mathcal G_2(p_{N+1})/\mathcal G_1(p_{N+1})\to0\), derived a rigorous chain-theorem lower bound for the numerator, and exactly verified \(2/9\le R_N\le5510/12
PARTIALIndependently verified \(h_3(4)=11\), \(h_3(5)\ge19\), and exact \(f(n)\) through \(n=18\); proved the Davies--Illingworth threshold/peeling template cannot beat constant \(2\), but did not improve th
PARTIALExact exhaustive rational enumeration proves the signed maxima \(0,0,2,4,6,8,12\) for \(1\leq n\leq7\), with a standalone checker; the asymptotic exponent remains in the verified interval \(1.77898\ld
PARTIALThe two page-cited ZFC constructions are verified to contain \(K_{\aleph_0}\); an ordered lower-clique theorem and exact truncation formulas are proved, while the general problem remains open at the c
PARTIALproved the sharp bound \(p-k!\geq\operatorname{nextprime}(k)^2\), certified equality solutions \(769\) and \(6\,227\,021\,089\), and exactly corrected/enumerated the strict sequence through \(10^9\) (
PARTIALmodulo Dirichlet every class is nonempty and infinitely many primes lie at or above each class; an independent exact certificate gives \(p_1,\ldots,p_{18}\), while exact-class infinitude and the behav
PARTIALverified g(4..14)=(3,0,3,3,3,3,3,3,4,4,4), extended Simonovits's Omega(n^(1/3)) bound to every sufficiently large order by Hajós joins, and constructed delta>=4 examples for every n>=56; the linear-gr
PARTIALverified the omitted stronger linear bound \(R(k+1,k)-R(k,k)\geq2k-2\) for \(k\geq5\), proved its optimality within the Xu--Shao--Radziszowski adjacent construction family, and reduced a proportional
PARTIALLive page status “OPEN - $100” (accessed 2026-07-27); “Currently working on this problem” lists CrashoverrideX and Sam_Petkov, so the mandatory no-collision rule requires no attempt.
SKIPPEDmodulo Guth–Maynard, \(13/30\le\liminf f\le1\) and \(\limsup f\ge13/30+\log2\); all three questions reduce exactly to \(f(p)\), and exact integer computation certifies the unique extrema through \(2{,
PARTIALExhaustively cross-checked all 171,551 connected 6-regular circulant parameter triples through order 80; every one of the 5,806 four-vertex-critical cases has a critical edge orbit, so no circulant ta
PARTIALUnder CH the explicit shift graph on \([\omega_2]^2\) has chromatic number \(\aleph_1\) and every lesser-order-type subgraph is countably chromatic; the first variant is blocked in the Foreman--Laver
PARTIALThe first question is relatively independent (even with GCH); the second has an exact positive solution under square+CH, hence in L, and reduces uniformly to the unresolved countable-chromatic-compact
PARTIALExhaustively verified \(f_6(6),\ldots,f_6(11)=15,15,23,27,35,39\), exhibited the unique 11-vertex 39-edge core, and reduced \(f_6(n)\sim n^2/4\) exactly to the two missing upper bounds (L5) and (I6);
PARTIALErdős #911 is exactly reduced, up to explicit constants, to proving \(\beta(d)=\omega(d)\) for bipartite minimum-degree-\(d\) targets; projective-plane incidence graphs certify that biclique extractio
PARTIALExact identities reduce #890 to a parity-level almost-prime tuple problem and a coefficient-1 maximal-order bound for \(\omega(\prod_{i<k}(n+i))\); an independently rechecked exhaustive scan on \(5{,}
PARTIALAn elementary Pell-orbit construction gives infinitely many n with five divisors in the required window for every C>24 (explicitly, the whole stated orbit works for C=25), improving the live comment's
PARTIALLive page status is OPEN, but “Currently working on this problem” lists MaskoMys and zitterbewegung (accessed 2026-07-27); mandatory no-collision stop.
SKIPPEDErdős–Rosenfeld, open at k=5: do five integers exist sharing at least five common factor-differences D(N)={|a−b| : N=ab}? The cases k=2,3,4 are solved. We ran GPT-5.6 Sol on it and found no witness — but a strong partial: a clean reformulation (a k=5 witness is exactly a 5×6 square-additive rectangle, equivalently a K₅,₅ biclique), an independent re-verification of the known k=3 (Guiduli) and k=4 (Bremner) examples, and three separate proofs that three different near-miss families are exactly saturated — Bremner's k=4 rectangle, Choudhry's brand-new June-2026 5×3 family, and a community list of 71 4×6 rectangles each admit no further row or column, all re-run on our side. Honest gap: no global obstruction proved; a witness could lie past every region searched.
PARTIALFULL WRITE-UPExact exhaustive verification proves the sharp threshold for every nonvacuous n<=36 and every required odd cycle length through C13; the uniform endpoint remains open and requires an arithmetic Hall-e
PARTIALThe live page status is OPEN with 0 claimed proofs, but it lists Svyable as “Currently working on this problem” (and its discussion contains Svyable’s unverified 3 May 2026 complete-answer post), so t
SKIPPEDProved that \(3+2\sqrt2\) is the sharp exponent for every block-size schedule certified by the posted \(C_k\) carry-closure lemma; the original range \(1\le c<3+2\sqrt2\) remains open.
PARTIALProved the elementary uniform bound \(F(A,X,k)\ll_k X^{\Delta_k}\log X\) with explicit \(\Delta_k\downarrow1/4\), upgraded it to the literal live-page inequality for every \(\epsilon>1/4\), and exactl
PARTIALThe authoritative live page (accessed 2026-07-27; status OPEN; 0 claimed proofs) lists current workers old-bielefelder, pommeret, and joshoakley, so the mandatory non-collision rule requires no attemp
SKIPPEDThe live page status is OPEN with 0 claimed proofs, but Woett is listed under “Currently working on this problem”; no attempt was made.
SKIPPEDLIVE PAGE status OPEN; mandatory gate triggered because it lists 1 claimed proof—a partial proof claimed by Samuel Korsky (using GPT 5.6-Pro), submitted 2026-07-26—and lists “Currently working on this
SKIPPEDThe authoritative live page is OPEN and shows “Currently working on this problem: None”, but it lists 1 claimed proof—a full proof claimed by RayYoung, Keheng Zhu, and Yanping Luo, submitted 2026-07-1
SKIPPEDexplicit 447-in-3159 admissible/CRT certificate and exact fixed-width scan through x=10^8 verified; the live large-x,large-y question remains open because the missing step is simultaneous primality of
PARTIALProved (modulo PNT and Mertens) liminf F(k)/(k log log k) >= e^gamma and Q(k) >= (e^gamma/2-o(1))k log log k; independently verified exact F(k) through k=13, full gap sets through k=8, and F(100)>=482
PARTIALCertified every even consecutive-prime gap through 1572, proved \(r(92,489,057,256,472,962,493)\ge1574\), exactly computed \(r(50,847,533)=254\) with all prefix breakpoints, and reduced the two open l
PARTIALVerified the repaired \(h(x)\gg(\log x)^{1/3}\) and \(h(x)\ll_\varepsilon x^{0.41+\varepsilon}\) reductions, computed the exact frontier through 105,000,000 starts, and found \(R(14)=R(15)=19,205\); t
PARTIALelementary reduction and an independent exact scan exclude y<=10^9; modulo Hercher's published exhaustive Theorem 1, all y<1.4e12 are excluded, but the unbounded problem remains open.
PARTIALExact modulo the published fixed-pair theorem, every \(\binom n2\) with \(4\leq n\leq10^{32}\) has half-triangle multiplicity at most 4 (only \(n=78\) attains 4); a 4.319-billion-candidate exact sieve
PARTIALProved \(A_3\cap[0,2/9)=\{0\}\), certified the exact five-vertex Lagrangian gap at \((189+15\sqrt5)/961\), and isolated the unproved endpoint-jump lemma hidden by the published \(0.2316\) rounding.
PARTIALLive page status is OPEN with 0 formal proof claims and no current worker, but its discussion page (accessed 2026-07-27) says “GPT-5.5 Pro claims a solution to the 2nd question here,” so the mandatory
SKIPPEDProved an elementary complement/extension equivalence and a hereditary LS(4,5,k+5) obstruction, independently verified the k=4,6 exceptional cases, and isolated k=16 at the open LS(4,5,21) barrier; pr
PARTIALexact equation-only A(10^8)=236 (231 distinct pairs plus 5 perfect diagonals), with a dual-engine checker through 10^7, endpoint re-factorization through 10^8, an exact aliquot-cycle reduction, and a
PARTIALVerified \(h(10^6)=6{,}437{,}809\), an explicit 46-member coprime sigma-fiber, and the sharper named-theorem bound \(h(x)>x^\alpha\) for every \(\alpha<1.4313775065\ldots\); exponent \(2\) still requi
PARTIALproved an exact exhaustive range theorem through \(10^6\), certified \(g(2^{21}3^{12}5^6)=26{,}819{,}695\) by two exact counts, and isolated the standard route's missing smooth-shifted-prime theorem;
PARTIALThe authoritative live page is OPEN but lists “1 claimed proof for this problem”; per the mandatory collision rule, no mathematical attempt was made.
SKIPPEDProved the fixed-window transfer \(R(n+h)-R(n)\geq r(h+1,n+h)-1\); with the published \(r(4,t)\) bound, superquadratic gaps occur in every three consecutive positions (lower density at least \(1/3\)),
PARTIALProved an exact rainbow-triangle identity and sharp t=2 minimum, proved problem 811 positive for forests and graphs with 2-core K3, proved lexicographic amplification always fails for C6, and reduced
PARTIALExhaustive certificates prove computationally that \(M(n)=0,1,3,5,7,11,14,17,23,30\) for \(1\le n\le10\), but the uniform dense construction/colored-completion lemma needed for Erdős #810 remains open
PARTIALThe live problem remains open only for C7; exact computation and an explicit certified witness give chi_S(10,26,C7)=13, while the uniform high-density path-length lemma needed for the n→∞ asymptotic i
PARTIALexact \(K_4\)-free edge/independence frontiers through \(n=11\) prove the sharp finite bound \(\alpha\ge n\ln t/(t\ln4)\), and the open asymptotic case is reduced to a quantified complete-tripartite-r
PARTIALThe literal zero-allowed function is exactly the standard nonzero function shifted by one; this gives exact values through the stated small regime, and a reproducible UNSAT computation gives \(S_+(13)
PARTIALindependently verified the exact table g(n) for every 0<=n<=65; #791 remains open between the Yu and Kohonen asymptotic constants.
PARTIALExact, from-scratch computer-certified values are \(g(n)=1,2,3,4\) on \(n=1\!-\!3,4\!-\!7,8\!-\!13,14\!-\!19\), respectively; the asymptotic problem remains open.
PARTIALFor the distinct-elements problem, proved computationally that \(M(20)=13\), recomputed the exact table \(M(N)\) for \(2\le N\le25\), and isolated the one-rank signed-relation defect that blocks the p
PARTIALVerified a 5-QP(4) of positive squares, correcting the 2003 “no 5-QP(5) found” observation; exhaustively proved diameter at least 4 for five terms with least gap at most \(10^8\), while both arbitrary
PARTIALproved the exact Cramér-window/tail decomposition, APRCL-reverified witnesses for 2<=n<=100, and BPSW-checked (not proved) the n=3001 candidate; the uniform prime-gap/prime-pair step remains open.
PARTIALLive page status OPEN; it lists 1 claimed proof (a partial proof claimed by Jeffrey Zeng, submitted 2026-07-24 17:13:29) and no current worker, so the mandatory no-duplication rule applies.
SKIPPEDLive page status is OPEN (accessed 2026-07-27), but it lists “1 claimed proof for this problem”—a partial proof claimed by Jeffrey Zeng and submitted 2026-07-24—while “Currently working on this proble
SKIPPEDThe general problem remains open, but the two live questions are rigorously equivalent, and an independently checked exhaustive computation classifies every graph through seven vertices: exactly \(K_7
PARTIALThe open problem is not solved; an independently rerun exhaustive census proves the exact values of \(M_{13}(T)\) for every 6- and 7-vertex tree, with explicit 4-chromatic graph6 witnesses and a preci
PARTIALexact f(n)=(3,3,4,4,6,6,6,6,6) for 3<=n<=11 with a from-scratch exhaustive verifier; proved an Omega(n^(3/4)) projective-truncation obstruction and isolated a quantifier gap in the current conditional
PARTIALThe live page is OPEN but lists 1 claimed proof—a full proof claimed by Eric Li (using GPT-5.5 Pro), submitted 2026-07-17—and lists “Currently working on this problem: None”; the claimed-proof stop co
SKIPPEDThe live page is OPEN but lists 1 claimed proof—a full proof claimed by Liam Price (using GPT Pro), submitted 2026-07-15—and lists “Currently working on this problem: None”; the claimed-proof stop con
SKIPPEDcorroborated Lu's stronger \(f(n)\geq n^{10/143}-2\) literature bound, independently verified nine MOLS(108), and exhaustively ruled out an eleventh row in the published matrix's natural equivariant f
PARTIALThe unrestricted problem remains open; verified here are an explicit Hall-multiplier proof excluding cyclic order 12 and a from-scratch exact exclusion of every non-prime-power cyclic order through 10
PARTIALFor the intended range r>=2, proved the conjecture uniformly for n<=r+3 and exactly for (r,n)=(3,7), with a standalone exhaustive checker; the general covering-design inequality (7) remains open.
PARTIALLive page status is OPEN with 0 claimed proofs, but it lists “Currently working on this problem: skominers” (accessed 2026-07-27); the mandated collision rule therefore forbids an attempt.
SKIPPEDLive page status is “OPEN - $78”, with SkyYang listed under “Currently working on this problem”; stopped to avoid colliding with a current worker.
SKIPPED\((1/e-o(1))\log n\le f_{709}(n)\le7\lceil n^{3/7}\rceil\), with a direct checked Hall certificate \(f_{709}(999999)>4.151303\); the asymptotic order remains open.
PROVEDThe polynomial upper bound remains open; this run gives explicit checked clique certificates including \(L(5)\ge12,L(7)\ge16,L(8)\ge19\), and proves by a standalone exhaustive checker that the triangu
PARTIALLive page status is OPEN, with 0 claimed proofs, but pollodumas is listed under “Currently working on this problem” (checked 2026-07-27), so the mandatory no-collision gate requires stopping.
SKIPPEDClassical finite-lcm reduction verified; a density-preserving blow-up gives a primitive gcd-1 reciprocal-divergent example of exact density 2/3, and all 1024 five-prime semiprime families were exhaust
PARTIALLive page status OPEN; it lists “1 claimed proof for this problem,” triggering the mandatory no-duplication stop.
SKIPPEDProved the exact Jacobsthal-quotient reduction, the unconditional bound \(\limsup\epsilon_n\le e^{-1}\), and a from-scratch exact table \(L(n)\) for every \(11\le n\le70\); the missing uniform linear
PARTIALThe live page status is OPEN, but it lists 1 partial claimed proof (Jake Weatherhead, submitted 2026-07-14) and “Currently working on this problem: None”; stopped under the mandatory claimed-proof col
SKIPPEDproved an exact LCM-seed valuation reduction, certified f(lcm(1..150)-1)=924 and the n<=100000 record table, and found a rigorous counterexample to the Fourier-tail lemma underlying arXiv:2604.23784's
PARTIALReduced #683 exactly to a fixed-power saving for the gap function f(y), proved c=1/10 for every k<=2,845,920 modulo Nagura and Nair--Shorey with all finite exceptions checked, and exhaustively verifie
PARTIALLive Erdős Problems #681 page status is OPEN, but it lists “Currently working on this problem: stellachenyl” (accessed 2026-07-27), so the mandatory no-collision rule applies.
SKIPPEDExact reduction obtained; the first assertion holds for every \(116\le n\le10^9\) (next-prime witness), with exactly seven smaller exceptions, while the uniform square-root gap step and the exponentia
PARTIALLive page status OPEN; mandatory collision guard triggered because it lists 1 claimed proof (a partial proof claimed by Liam Price using GPT 5.6 Sol Pro, submitted 2026-07-27 13:14:00); “Currently wor
SKIPPEDModulo verified square/cube/fifth-power theorems, any counterexample at the first unresolved lengths k=35..38 has prime exponent at least 7 and largest term at least 13,357,343; 13,750,206; 47,045,881
PARTIALExact finite progress—proved by a standalone exact certificate that for every d>=3 the minimum five-point diameter is 10+t_*=10.097819085934546..., with an explicit R^3 Gram construction; the fixed-d
PARTIALProved the uniform bound c(p,q) <= (p-2)/(C(p,2)-q), the neighborhood recurrence, and two infinite families of strict adjacent jumps; reduced the remaining p=4 jump exactly to the independence exponen
PARTIALExact catalogue-backed verification proves the \(k=4\) assertion for every graph on at most 12 vertices, with sharp threshold \(\chi=5\), and determines the unique smallest path-version witness at ord
PARTIALThe live page status is OPEN with a “Partial Solution” notice stating “A claimed solution has been posted in the comments”; the discussion page shows an April 2026 claimed counterexample/proof thread,
SKIPPEDProved that any total-\(\le39\) obstruction reduces to one on a palette \(11\le q\le22\), and computationally certified with two independent solvers that every 11-color obstruction whose unoriented un
PARTIALIndependently reproduced the known double-critical census through 12 vertices and certified that any order-13 noncomplete double-critical graph must be 6- or 7-chromatic; the general Erdős–Lovász Tiha
PARTIALExact, reproducibly checked values are \(f(n)=1\) for \(1\le n\le4\), \(f(n)=3/2\) for \(5\le n\le10\), and \(f(n)=2\) for \(11\le n\le16\); the asymptotic limit remains open.
PARTIALreduced both limits exactly to growth rates of N_k(m), proved the m=3 exponent 1/2 modulo Kim, improved liminf g_4(n)/log n to 1/(6 log 2) modulo Morgenstern, and independently verified a 66-vertex gi
PARTIALStatus OPEN ($1000), but the live-site search index lists “Currently working on this problem | zitterbewegung” for #625 (cross-checked on the indexed zitterbewegung profile); direct Bright Data access
SKIPPEDf(n,k) is the fewest edges forcing every (k+2)-set of vertices to induce a subgraph with a vertex of degree ≥k. Round 1 reduced it to a finite-family Turán problem and flagged the general formula as unproven. Round 2 pushed on that gap — and mostly settled it, except for one claim we caught overreaching: it silently treated the 50-year-open Erdős–Simonovits even-cycle conjecture as solved. Corrected, and the corrected version is the more interesting finding: for most k, a clean formula is provably at least as hard as that named open problem — that's WHY it doesn't exist, not just that nobody's found it yet. Round 3 maps the regime Round 2 left untouched — the near-diagonal one (n−k small): the actual 1996 Erdős–Reid–Schelp–Staton theorem and Pikhurko–Thomason's 2002 follow-up, which disproves the ERSS conjecture, gives the k→∞ asymptotic, and proves the naive split-graph fails off-diagonal — plus an unconditional f(n,k)=C(n,2)−o(n²) for every fixed k. The fixed-k orders Round 2 already proved (k=2,3,…) re-derive here; they're confirmation, not new ground.
LIVEFULL WRITE-UPLive status is OPEN with no worker or claimed proof; primary sources leave \(2^{\Omega(\sqrt{\log n})}\le f(n)\le O(n^{3/2}2^{n/2})\), while a proof-carrying exhaustive computation establishes the exa
PARTIALproved \(\omega_1^2\to(\omega_1\omega,G)^2\) for every finite \(K_4\)-free block graph and reduced the full finite question exactly to 2-connected \(K_4\)-free blocks; the first unresolved-by-this-met
PARTIALClassified every finite target in the full \(G_1=C_4\) row—exactly the triangle-free, \(C_4\)-free graphs containing \(P_4\)—and found/verified the minimum 9-vertex, 11-edge \(C_4\)-free red-blue Rams
PARTIALThe live page status is OPEN, but it lists 1 claimed proof—a full proof by Cambie Stijn and Freschi Andrea (arXiv:2606.11174, submitted 2026-07-25)—and its newest comment says the solution is availabl
SKIPPEDProved the explicit 2-core reduction \(R(G,H)\le R(G,C_2(H))+(v(G)-1)(v(H)-1)\), reduced #568 to connected minimum-degree-two targets, and exhaustively isolated \(K_4^*\) as the unique order-at-most-f
PARTIALProved exact all-n formulas for R(H5,nK2), R(K3,3,nK2), and R(Q3,nK2), with an exhaustive standalone verifier; the full arbitrary-target problem remains open.
PARTIALProved \(R(K_4^*,nK_2)=2n+2\) for every \(n\ge2\) and computationally determined \(R(K_4^*,H)\) for every no-isolate \(H\) with at most four edges; the uniform \((2,3)\)-sparse problem remains open.
PARTIALproved a uniform barrier excluding every fixed pointwise two-colour merge of the standard four-colour stepping-up construction, and independently verified the known explicit 87-vertex colouring \(R_3(
PARTIALProved the exact piecewise formula R_alpha(4)=18,10,6, computed the complete n<=7 extremal table, and reduced the asymptotic question to existence of lim_k log R_alpha(k)/k, whose alpha=0 case is the
PARTIALverified a compact explicit 87-vertex coloring proving the known bound \(R_3(5)\geq88\), proved a uniform no-go theorem for the naive tournament stepping-up lift, and isolated the still-missing \(R_3(
PARTIALLive page status OPEN; it lists old-bielefelder under “Currently working on this problem” (and 0 claimed proofs), so the required collision rule forbids an attempt.
SKIPPEDindependently certified \(\hat R(K_{2,2})=15\) and uniquely \(K_6\) as the 15-edge host (up to isolates), proved \(45\le\hat R(K_{3,3})\le153\), but the conjectured diagonal factor \(n\) remains block
PARTIALAn explicit cubic-residue colouring and an elementary spectral count prove \(20\le R_3(K_{2,3})\le22\); unrestricted \(K_{20}\)/\(K_{21}\) feasibility is the exact remaining finite question, while cyc
PARTIALverified \(R_q(T)\leq q(n-2)+2\) for all sufficiently large fixed-bounded-degree trees (modulo Pokrovskiy Theorem 1.16), certified every \(R_3(T)\) for \(|T|\leq5\), and isolated the unresolved all-tr
PARTIALThe live statement is false at n=3 (R_3(C_3)=17>9); for the intended n>=4 problem, R_3(C_4)=11 and the sharp Toeplitz K32/K33 C_9 transition are verified, while unrestricted R_3(C_9)<=33 remains open.
FOUNDThe live page status is OPEN and lists RobSneiderman under “Currently working on this problem” (with 0 claimed proofs), so the mandatory no-collision rule requires stopping.
SKIPPEDThe live problem is open with no claimant or worker; current bounds miss by a \(k^{k/\ell}\) factor, and the report proves a precise colour-index-gap reduction plus fully checked finite certificates,
WALLCan two disjoint blocks of consecutive integers, both length >3, ever have equal products? 8·9·⋯·14 = 63·64·65·66 = 17,297,280 is the only known example. Round 1 established fixed-pair finiteness. Round 2: a real corollary narrows it further (fixed length RATIO is also finite, so any infinite family needs infinitely many distinct ratios), a 2024 paper's computer search conjectures this IS the complete classification, and a search reaching 10 million found nothing else across 1.4 billion comparisons. Round 3: a follow-up wrote out the actual proof of the fixed-unequal-length finiteness Round 2 had only cited — a clean Kulkarni–Sury case-elimination we verified by exact computation — and caught a subtle error in the naive root-comparison argument. It proves no uniform bound, so the full problem stays open; everything below the uniformity step is now settled.
LIVEFULL WRITE-UP#126 asks whether f(n)/log n → ∞, where f(n) is the fewest distinct primes that can support all pairwise sums of an n-element set — equivalently N(s)=exp(o(s)), N(s) being the largest set whose pair sums use ≤s primes. The window (½+o(1))s·log s ≤ N(s) ≤ 2^s has been open since Erdős–Turán. We didn't prove it and found no counterexample — but a strong partial: the conjecture reduces cleanly to ONE missing theorem (a uniform subexponential bound for binary S-unit equations over ℚ), the key extra structure is identified (the problem needs a clique of mutually compatible S-unit solutions, not the star a single binary equation gives), and the old local prime-by-prime route is provably dead. N(2)=4 exactly, plus explicit f(14)≤8 and f(21)≤12 — all verified on our side.
PARTIALFULL WRITE-UPcertified 247 distinct cycle sets on 11 vertices, so f(11)>=247; reproduced exact f(1..10), but the required unbounded-factor lower bound remains open.
PARTIALLive page status OPEN; collision rule triggered because “Currently working on this problem” lists Svyable (checked 2026-07-27).
SKIPPEDLive Erdős Problems #81 status is OPEN with 0 claimed proofs and “Currently working on this problem: None,” but it lists jtraverso under “I am working on formalising the results on this problem”; this
SKIPPEDLive page status OPEN; it lists “Currently working on this problem: memeister27,” so no attempt was made under the required collision rule.
SKIPPEDThe live page status is “OPEN - $500” with 0 claimed proofs, but it lists Mahmudsudo and quintessen as “Currently working on this problem”; stopped before any mathematical attempt to avoid colliding w
SKIPPEDExhaustively verified the exact complete-bipartite minimum and uniqueness for every \(4\le n\le10\) and \(2\le d\le\lfloor n/2\rfloor\); the unresolved step is a uniform fixed-small-\(d\), all-\(n\) s
PARTIALProved the common-4-chromatic conclusion for every pair of canonical finite-order shift-graph ages and computed exact first thresholds 4, 9, 21, 85, 8574 through order five; the arbitrary good-family
PARTIALExplicit induced-C6-free lexicographic powers have 17^k vertices and homogeneous number 3^k, giving c_EH(C6) <= log_17(3), with T_C6(4)=18 and an exact 104-type six-vertex threshold table; the uniform
PARTIALExhaustively verified the new exact cases \(h_{C_4}(n,1)=2\) for every \(14\le n\le18\), with explicit two-cycle witnesses and a 55-second standalone checker; the uniform \(\Omega(\sqrt n)\) problem r
PARTIALProved, modulo Schoenberg/Mertens/Bertrand, a dense continuum-sized family of irrational points generated by an explicit recursion with infinite upper-right Dini derivative; the possibility of a finit
PARTIAL#41 asks whether every infinite B₃-set (all triple sums distinct) must have its normalized counting function dip to zero: liminf A(x)/x^(1/3)=0. It's known for every even order and open for odd. GPT-5.6 Sol found no resolution — but a strong partial: (1) the exact reason the classical B₂ method breaks at B₃, a parity wall (a B_h condition controls an h-variable signed form, which can balance + and − only when h is even); (2) a real conditional theorem — any B₃-set with positive critical density must have logarithmic gap spikes, so #41 holds for every smooth-gap B₃-set; (3) the sharp finite bounds 1 ≤ liminf R₃(N)/N^(1/3) ≤ limsup ≤ 1.51546…; (4) an explicit construction hitting the critical exponent 1/3 at checkpoints (but with liminf zero). Constants and citations re-verified on our side.
PARTIALFULL WRITE-UPPliego's theorem rules out every g with a positive polynomial lower bound, an exact compactness reduction isolates the remaining subpolynomial frontier, and a verified 25-term cap-3 witness gives psi(
PARTIALexact finite square-completion numbers are verified through target 101, with a rigorous block reduction isolating the missing uniform \(q(N)<(2-o(1))\sqrt N\) lemma; the live limsup constant remains o
PARTIALExact fresh-shell minima were certified at 13 values through N=128, and two natural independent-scale approaches were rigorously shown to retain the log-squared barrier; the missing step is a correlat
PARTIALProved that bounded pairwise gcds force an explicit natural density, so every counterexample must have unbounded common factors; verified 27,567 finite systems and an exact nonsummable gcd-2 example.
PARTIALThe authoritative live page is OPEN - $1000 but lists sproutseeds, Rynaldos, and Lumantis under “Currently working on this problem” (accessed 2026-07-27), so the mandatory no-collision rule applies.
SKIPPEDThe live page is OPEN - $250 but lists 1 claimed proof (a partial proof claimed by Liam Price, submitted 2026-07-24, asserting the first question) and Sam_Petkov as currently working on the problem; m
SKIPPEDProved an elementary cylinder-budget theorem forcing convergence for the regular oriented-CRT block template used in the live comments, and exactly computed the finite harmonic optima M(N)=3/2 through
PARTIALLive page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists lof310 (checked 2026-07-27), so the required no-collision rule applies.
SKIPPEDLive page status “VERIFIABLE - $25”; current workers are listed as duckmerc and Aurelien_Col, so the mandatory no-collision rule forbids an attempt.
SKIPPEDCounterexamples to eventual strictness found and checked. Unresolved.
PARTIALLiterature map plus several open cases — no resolution.
NO PROGRESSOpen — a related variant was solved by others (Joret–Micek–Reed–Smid) while we worked it.
NO PROGRESSThe first located open case is reduced to an explicit 106-vertex 2-connected kernel with verified expansion/degree/clique constraints, and Nikiforov Case 4.1 has a reproducible factor-two proof gap; n
PARTIALPage status: OPEN; nevertheless, erdosproblems.com lists #550 under “Solution Claims” (Eric Li claims a full proof in arXiv:2606.23659), so the mandatory claimed-proof stop condition applies.
SKIPPEDBright Data reached Erdős Problems #547 but the origin returned HTTP 522; the latest accessible indexed page and discussion-thread representations (crawled three weeks to one month ago) report status
SKIPPEDproved from scratch that the eight-edge colex graph has Ramsey number 18 (so its exact matching counterexample gap is 5), and located the missed 1989 theorem \(R(\widehat K_{n,2})=R(K_n)\) for every \
PARTIALAn exact checker exhibits nested critical graphs through k=8, proves that the unique (3,9;35) graph has independent-8-set transversal number 9 so the nesting cannot continue at k=9, and proves a sharp
PARTIALVerified \(F(1)=1\), \(F(2)=9\), and certificate-backed \(F(3)=54\); an explicit checked colouring proves \(F(4)\geq201\), while the asymptotic tower-versus-double-exponential gap remains open.
PARTIALThe order is already Θ(√N) — the refinement is the question. Open.
NO PROGRESSSieve machinery explored; the full problem appears untouched. Open.
NO PROGRESSExplicitly reverified an 8-coloring of [1,5362], derived a checked infinite lower-bound subsequence, repaired the printed width-111 template, and proved the width-380 tail constant 148 locally maximal
PARTIALproved the sharp identity A(p^2-p+1)=p-1 for every prime p and independently verified A(n)>=227 on 10^6<=n<=10^7; the uniform limit remains blocked by a worst-case rough-semiprime gap lemma.
PARTIALThe stronger composite-only inequality with C=1 is certified (>0.961) for every real x in [10^6,10^9], and the remaining all-x asymptotic step is the stated uniform balanced-semiprime Type-II lower bo
PARTIALExplicit CRT arithmetic gives f(n,500)=218 (hence any uniform constant is at most 0.436), and exhaustive periodic enumeration gives the exact minima for every t<=13; Erdős #461 remains open.
PARTIALReduced the conjecture to a precise growing paired-gap window and certified every prefix record through \(n=10,000,000\), with maximum \(148\) first at \(n=2,886,673\); unboundedness remains open at t
PARTIALHow long a run of consecutive integers can stay above the normal prime-factor count ω(n) > log log n? Round 1 gave the explicit CRT lower-bound construction and reported the honest bad news: no non-trivial upper bound existed anywhere. Round 2, same day: a second model built a real one — a genuine sieve-theoretic upper bound miles below the trivial prime-gap bound, with a fully checkable 5-integer example. Half its six background literature citations didn't survive verification and are dropped below; the actual proof — checked independently, not just accepted — doesn't depend on any of them.
LIVEFULL WRITE-UPThe problem remains open; verified current progress is Croot--Yip's July-2026 quadratic-intersection theorem, and this report adds an exhaustive sharp computation of the balanced all-prime-sum paramet
PARTIALAny hypothetical witness sequence is rigorously forced by a uniform Selberg prime-pair bound to have gaps at least $\exp(m^{c_A})$, an exact synchronized-chain reduction and two-witness maxima through
PARTIALRound 5 — the capstone. The 0-indexed entirely-complete boundary has a closed form, t0(α) = min over k≥0 of (2^k+1)/α^k, now triply confirmed (our exact-rational compute, a same-family skeptic, and a cross-family GPT-5 Pro proof whose key move is Graham's 1964 Lemma 4 in disguise). The honest payoff is provenance: under the reindex u=t/α the 0-indexed sequence IS the literature's sequence, so for t<α we sit inside Graham's 1964 unit square (Acta Arithmetica 10, primary-verified — Lemma 1 the greedy criterion, Theorem 5 the powers-of-two obstruction, Theorem 6 the regions whose pullback is exactly our fragmented complete set, Theorem 3 complete iff entirely complete there) and for t≥α inside van Doorn 2026. So the set, its ≈85% area, and its region structure are Graham's; the 1-indexed closed thresholds are van Doorn's. What is genuinely new is the independent cold re-derivation plus the one-line min-formula boundary of Graham's set in 0-indexed coordinates (explicit shelf + least-minimizer) — a unification not previously written — not a new theorem. We think that is the better, truer story.
LIVEFULL WRITE-UPSubset-sum checker built; the asymptotic threshold is unresolved.
NO PROGRESSProved the exact forward-obstruction characterisation and \(a_n\leq\binom n3+n\), independently recomputed 3,000 terms and the collision/tail split, but the required uniform \(m^{2+o(1)}\) bound remai
PARTIALproved the complete finite-prefix-shadow criterion for all ultimately periodic pairs, exhibited \(A_q=q\mathbb N\cup\{1\}, B_q=q\mathbb N\), and exhaustively verified the stated small-period tables; t
PARTIALLive page status OPEN, but it lists 1 claimed full proof—Will Sawin's submitted 2026-07-18 construction giving a positive-density counterexample to the second question (now arXiv:2607.15419)—so the ma
SKIPPEDLive page status is OPEN, but it lists 1 claimed proof (and no current worker); the mandatory collision rule therefore forbids an attempt.
SKIPPEDExact modular certificates prove the second inequality for every \(5\leq n\leq200000\) (and equality defeats it exactly at \(n=1,2,3,4\) in this range); the uniform prime-residue lemma and the first q
PARTIALNew computational contribution landed; infinitude still open unconditionally.
PARTIALLive page status is OPEN with 0 claimed proofs, but SamKorsky is listed under “Currently working on this problem” (accessed 2026-07-26), so the mandatory no-collision rule requires stopping without an
SKIPPEDUniform elementary equivalence proof, corrected rigorous 5/6 subsequence argument, and an exact independently checked table extended from N=25 to N=50; the asymptotic question remains open at the expl
PARTIALLive page status on 2026-07-26 is open/VERIFIABLE with 0 claimed proofs, but “Currently working on this problem” lists aditya, will0708, and Ary300; stopping to avoid colliding with current workers.
SKIPPEDLive-page status is OPEN (0 site-counted claimed proofs; “Currently working on this problem: None”), but the live discussion contains Yuren Tang’s explicit 19 June 2026 claim of an affirmative, sorry-
SKIPPEDexact exhaustive search proves N(b) <= 7 for b <= 1000, with equality exactly at 733, 739, 787, 839, 863, 898, and 907; the uniform O(log log b) question remains open, and the precise common-denominat
PARTIALLive page status OPEN; it lists 1 claimed proof (partial, attributed to Robert Schuh using GPT 5.6 and Kimi 2.6, submitted 2026-07-20 by 15Redstones) and CURRENTLY WORKING markers for Woett and Quanyu
SKIPPEDThe live Erdős Problems page marks #301 OPEN with 0 claimed proofs, but lists Woett and Quanyu_Tang under “Currently working on this problem”; per the collision rule, no attempt was made.
SKIPPEDexact v(1..7) = 2,2,4,11,17,103,733 and the independently verified bracket 2307 <= v(8) <= 27539; the asymptotic problem remains open.
PARTIALThe open half is reduced exactly to a growing-prime leading-digit avoidance problem, and two independent exhaustive sieves certify 10,323,214 gcd-\(1\) values through \(10^8\); infinitude remains unpr
PARTIALLive page status is OPEN with 0 claimed proofs, but jgold is marked “Currently working on this problem”; stopped under the mandatory collision rule (accessed 2026-07-26).
SKIPPEDThe live page status is OPEN, but it lists Rafikzeraoulia2025 and Ritvik_Nayak as currently working on Erdős problem #288, so the mandatory no-collision rule applies.
SKIPPEDLive page status on 2026-07-26 is FALSIFIABLE/open with 0 claimed proofs, but it lists stffp under “Currently working on this problem” (and “Interested in collaborating”); mandatory collision-avoidanc
SKIPPEDExact distinct-odd greedy termination verified for all 202,660 reduced fractions a/b with odd b<=999 (at most 27 terms), plus a rigorous reduction to the below-2/3 convention; uniform termination rema
PARTIALRigorous reduction to boundedness of H_k, FGKMT-based sublinear bound, repaired density-one proof, and independently verified exact frontier H_3(172)=76; the required uniform O(1) bound remains open.
PARTIALPositive lower density is proved for every Pisot or Salem base C (rigorous modulo Ballot–Luca), while arbitrary real C still requires the pointwise Romanoff correlation bound (6.1).
PARTIALLive page status is FALSIFIABLE/open with 0 claimed proofs, but it lists current workers jgold, alansbor, Bradford, and auro (accessed 2026-07-26), so the mandatory no-collision rule applies.
SKIPPEDExhaustive dual-algorithm computation gives exact R_D(10^j) tables through 10^9 and sharp finite-range constants, while the open asymptotic is reduced to logarithmically long holes in the short-gap in
PARTIALExact exhaustive computation proves max_{n<=10^9} f(n)=19 with four maximizers and a reproducible checker; the open asymptotic reduces to a growing-dimension uniform prime-tuple sieve bound whose fact
PARTIALLive page status is OPEN, but it lists aaryanium under “Currently working on this problem” (checked 2026-07-26), so the mandatory no-collision rule requires stopping.
SKIPPEDExact dual-sieve verification proves \(C_N\le2.23N(\log N)^2\) for every \(10\le N\le10^7\) (and \(2.15\) for \(100\le N\le10^7\)); asymptotically the unresolved input is the large-gap tail (8), while
PARTIALproved the quantitative tie bound \(E_N\ll N/\sqrt{\log N}\) modulo the standard Selberg sieve, reduced both density claims exactly to \(S_N=o(N)\), isolated a fixed admissible triple sufficient for i
PARTIALThe requested limiting ratio was never proved. Open.
NO PROGRESSThe sharp scale-local constants are \(\Delta_2,\ldots,\Delta_6=2,4,11,44,226\), with explicit checked periodic witnesses; a logarithmic lower bound for #187 would follow from the still-missing uniform
PARTIALverified a five-class near-colouring of PG(7,2) with exactly one monochromatic Fano plane and proved by a standalone exhaustive checker that every proper five-colouring is at fixed-label Hamming dista
PARTIALIndependently verified R(Q1)=2, R(Q2)=6, and R(Q3)=13 (explicit K12 witness, 8,063 orbit reproduction, exhaustive zero at K13); Erdős 181 remains open, with D(n)=O(2^n) the precise sufficient density-
PARTIALReduced the problem exactly to finite weak hypergraph colouring, proved R(2,2)=8, computationally certified R(3,2)=100, and supplied a solver-free-verified 2-colouring of [5000] showing R(2,3)>5000; n
PARTIALExhaustively verified the exact inverse-Ramsey and maximum/average independent-set tables for every triangle-free graph through 13 vertices (unique minimum ratio \(244/169>4/3\)), isolated the equival
PARTIALThe live #162 definition is ill-posed—every integer k>n is vacuously admissible, while restricting k≤n gives F(n,α)=n eventually for every α<1/2 and no admissible k at α=1/2.
FOUNDproved the exact closed form for all n=t+2 (including literal two-jump examples t=4,n=6 and t=5,n=7), verified it exhaustively, and isolated hypergraph Nikiforov as the current asymptotic wall.
PARTIALexact exhaustive recomputation gives F(1..7) = 1,0,1,6,72,2320,245765 with a no-duplicate k=7 certificate; the corrected Konyagin identity is proved, but the asymptotic log-factor gap remains open.
PARTIALErdős's largest bounty ($10,000): an actual asymptotic formula for r_k(N), the size of the largest subset of {1,...,N} with no non-trivial k-term arithmetic progression. "Probably unattackable," Erdős said — and it still is. We sent two frontier models after it independently, expecting no proof, and got no proof — but both surfaced and verified two genuine 2024–2026 breakthroughs most casual summaries (including our own brief) had gotten stale, plus a precise account of exactly why "formula" is a different kind of hard than "bounds."
NO PROOF TODAYFULL WRITE-UPVerified the published CPAP-10 from embedded ECPP certificates, computationally verified that no CPAP-7 ends by 10^9, and reduced the general case conditionally to explicit admissible linear forms; no
PARTIALLive page status OPEN, but it lists 1 claimed proof and Lherdos as currently working on the problem; mandatory no-duplication/no-collision stop.
SKIPPEDUnder the live page's literal definition, \(R(n;3,2)>\lfloor1.022^n\rfloor\) for every \(n\ge500\), and EKR strengthens this to \(R(n;3,2)\ge2^{(1/4-o(1))n}\); hence the requested \(C^{\sqrt n}\) uppe
PROVEDThe universal limit is equivalent to Erdős's finite-density formulation, and the EHS ordered 2-shift is proved to have sharp \(\Theta(n^{3/2})\) defect; exact DP gives \(\beta(S_{12})=40\).
PARTIALIndependently proved and exhaustively checked the exact thresholds \(F(2,r)=2\) and \(F(3,r)=r\) for odd \(r\), \(r+1\) for even \(r\), and isolated the density-free short-cycle-transversal versus col
PARTIALproved the first inequality for every epsilon > 0.627760438360797, reduced the k=5 lower bound 43 to 4,195 checked critical candidates, and certified R(K_2 join C_5) >= 26; both full asymptotic questi
PARTIALExact post-1601 Paley values (including \(\omega(P_{1669})=\alpha(P_{1669})=13\)) and the entropy-\(\log m+O(1)\) reduction are verified, but the required uniform strongly explicit family remains open
PARTIALLive page status “OPEN - $250”; the “Currently working on this problem” marker names Sam_Petkov (fetched via Bright Data on 2026-07-26), so the mandatory non-collision rule applies.
SKIPPEDLive page status “OPEN” with “0 claimed proofs” and no current worker, but its live discussion contains Przemek Chojecki’s explicit 12 Apr 2026 claim of a proof of the first statement; the mandatory c
SKIPPEDlive page is OPEN with no worker/claim; verified \(n\leq3\) for all countable \(\beta\) and, modulo Jones 2018, every finite \(n\) for \(\beta\leq\omega\cdot2+1\), leaving \(n\geq4,\ \beta\geq\omega\c
PARTIALLive page status OPEN; it lists one claimed partial proof (Liam Price using GPT Pro, submitted 2026-07-20, claiming the lower bound 0.38055470) and names JJ_ as currently working, so the mandatory non
SKIPPEDThe live page status is OPEN - $1000, but “Currently working on this problem” lists Emmanuel_Audigé_Y., Rynaldos, TerenceTao, and KMendoza (accessed 2026-07-26), so the mandatory no-collision rule app
SKIPPEDreduced the documented \(n=13\) frontier exactly to \(m=33,\ldots,54\) and supplied independently verified reduced witnesses with exact chromatic index in all 22 nonvacuous buckets; the universal colo
PARTIALproved an exact transition/frontier reduction and independently verified all cluster primes through 10^8 (202,208 total), but infinitude remains blocked already by the unproved H_1<=6/Maillet-level lo
PARTIALexact \(F(x)\) extrema through \(10^8\), exact short-interval parity tables, a rigorous Abel/correlation reduction, and a counterexample to anti-concentration as a sufficient intermediate target; the
PARTIALexact finite exception minima E(N) are exhaustively verified for every 1<=N<=80, with explicit witnesses and a 43,083,585-node from-scratch checker; the asymptotic problem remains open.
PARTIALexact exhaustive table through 1,117,175,146; proved computationally that 1,117,175,146 is the first integer requiring four powers, while the uniform boundedness question remains open.
PARTIALverified an explicit 420-digit member and infinite CRT family in \(A\), proved computationally that \(A\cap[1,10^9]=\{1,3\}\) with sharp exponent cap 28, and ruled out fixed local covering obstruction
PARTIALLive page status OPEN; “Currently working on this problem” lists alansbor, so no attempt was made to avoid colliding with a current worker.
SKIPPEDLive page status is OPEN - $5000 (accessed 2026-07-26), but it lists dbmcdonough under “Currently working on this problem” (0 claimed proofs; “Interested in collaborating”: None), so the mandatory no-
SKIPPEDLive page status `OPEN - $500`; it lists `Currently working on this problem: lbattu` (and `0 claimed proofs for this problem`), so the no-collision rule requires stopping before any mathematical attem
SKIPPEDFor closed A ⊆ [−1,1], does some continuous f have A as the cluster set of the Chebyshev–Lagrange interpolation sequence L_n f(x₀)? We first shipped this as an unflagged find; it wasn't. Przemek Chojecki posted the resolution to the erdosproblems.com comment thread on 30 April, Nat Sothanaphan confirmed it the same day, and Allen Hart announced a Lean formalization in early May, also confirmed. The top-level tracker badge (OPEN, 0 claimed proofs) just hadn't caught up — we mistook a stale badge for an unflagged result because our first pass never loaded the actual comment thread. Our contribution is a genuine one, just smaller than we first framed it: an independent from-scratch Lean rebuild, on our own compute, as a second confirmation.
CLOSEDFULL WRITE-UPerdosproblems.com's own statement of #1063 is just 'estimate n_k' — open-ended, and it stays open here. What we actually proved: n_k = exp(o(k)), a genuine improvement on Cambie's exp((1+o(1))k) bound, which fills one specific formalized open target (a Google DeepMind Lean placeholder). A reader correctly called out an earlier version of this entry for reading like a full solve with no checkable proof behind it — full corrected writeup and proof now public.
LIVEFULL WRITE-UPOpen — a recent Eric Li preprint claims more; we treat it as an external candidate, not ours.
NO PROGRESSStrong partial — the universal statement remains open.
PARTIALg₃(n): the largest A⊆[n] where every integer has at most two representations as a product of two elements of A. We first shipped Rishikesh Gajjala's construction as unflagged; it wasn't — his own proof-claims submission was already on the tracker since 15 July, and so was a separate, independent, earlier claim by Colin Snyder (Star Fleet Math) giving a sharper exact constant. Our from-scratch rebuild of Gajjala's proof still stands (with a real sorry found, traced, and confirmed harmless) — Snyder's competing claim is not yet independently checked by us.
CLOSEDFULL WRITE-UPDensity of integers where every prime factor has a divisor ≡1 mod itself — Eric Li's paper pins the exact constant c=1/(2√log2). Already independently formalized twice by different people (Li himself, and separately JohanLand). We're a third, from-scratch check, not the first.
CLOSEDFULL WRITE-UPNo counterexample, no general proof. Open.
WALLA $500-prize problem (#593, characterize the finite triple systems every uncountably-chromatic triple system must contain) plus a companion (#1177, three exact-cardinal questions about avoidance classes) — both already claimed by Eric Li, already flagged by the community as warranting extra scrutiny given his pace of claims. We gave it that scrutiny: cloned the repo, built from source, ran #print axioms on all 8 component theorems plus the unconditional master theorem. Clean.
CLOSEDFULL WRITE-UPStar Fleet Math (Colin Snyder, GPT-5.6 with a custom Lean harness) has quietly submitted full proof claims for four open Erdős problems on the tracker — ten days before we noticed. We almost shipped these as our own unflagged finds, on a triage brief whose search access was blocked from seeing the tracker's proof-claims pages. We caught it before posting anything: all four (#489, #394, #336, #1188) are already on record, zero comments, zero scrutiny. So instead of "we found this," here's the honest version — an independent from-scratch rebuild of two of the four, on our own compute, as a second opinion nobody had given yet.
CLOSEDFULL WRITE-UPRound 2: proofs for 3 and 4 primitive generators. General case open.
PARTIALDecidable up to the computed range, no counterexample — general case open.
PARTIALErdős and Graham call this 'almost certainly' true but 'beyond our ability' to prove. New reciprocal-exponent budget sharpens exactly what a counterexample would need to contain, and (conditional on a real distributed prime-gap computation) the conjecture holds for every gap through 10²⁰. First problem this session came from a fresh, self-directed scout rather than a shared transcript.
LIVEFULL WRITE-UPRemote-witness obstruction unresolved; a blind reattempt corroborated the architecture, no summable bound.
WALLF(n) counts iterations of n ↦ φ(n)+1 until a prime is hit. New elementary potential-function theorem pins the leading constant to 1 (down from 5/4) and resolves the prime-square case cleanly; with Rosser–Schoenfeld, F(n) ≤ √n + (7/4)loglog n + O(1). Exponent below 1/2, and the general fixed-prime-basin question, stay open.
LIVEFULL WRITE-UPFor which f does n+f(n) hit some output range far more often than chance, for every F growing slower than f? Confirmed: Maynard's 2016 clustering theorem really does build exact, verifiable collisions of n+τ(n); the real 1985/1997 Erdős–Pomerance–Sárközy mechanism is reconstructed correctly; and the naive τ generalization of it fails for a precise, provable algebraic reason. Round 5: a genuinely new result EPS never published — for every fixed m, infinitely many y have at least m representations y=n+τ(n), rigorous modulo a real 1985 theorem, independently reverified.
LIVEFULL WRITE-UPA reduction plus a conditional construction — no resolution. Open.
NO PROGRESSNo counterexample — ratio behavior verified computationally. Open.
NO PROGRESSGiven any finite Sidon set A, can you always extend it to a bigger one of near-(1−ε)√M density? A quantifier-correct construction reaches 1/√2, full density-1 for |A|≤2 (and |A|=3 via a named unpublished proof), and a proof that the obvious repair strategy (patch a perfect difference set) is structurally dead for seeds like Alexeev–Mixon's {1,2,4,8,13}. Round 2: the two |A|=4 patterns that provably embed in no Singer difference set turn out to embed cleanly in a different classical family instead — Singer failure wasn't the wall it looked like. Round 4: the round-3 obstruction sets do embed in Sidon spaces (rebuilt and independently verified, including a real density-value error caught along the way), but a cited nonexistence theorem shows the standard construction family is empty exactly where density-one completions would need to live. Round 5: cubic Bose–Chowla doesn't restore coverage either — a clean, verified structural negative. Round 6: two reasoners working independently converge exactly, and a fact Round 4 left unverified — zero binary critical Sidon spaces at dimension 5 — is now genuinely triple-checked.
LIVEFULL WRITE-UPNo counterexample found — search exhausted its budget. Problem open.
WALLDo sums of distinct factorials contain only finitely many perfect powers? The two-term case collapses exactly onto Brocard's equation — open since 1876. No proof, but a real quantitative bound, a fully reproduced computational sweep, and a concrete new angle on Brocard itself.
LIVEFULL WRITE-UPDo two disjoint blocks of consecutive integers with identical prime support only collide finitely often? A real structural transfer from our #677/#389 work, a sharper (partly-verified) counting bound, and a computational sweep we reproduced exactly — 16 for 16.
LIVEFULL WRITE-UPA real theorem for close prime triples, plus Maynard's sieve theorem, appears to unconditionally answer 'yes, infinitely many' to one of the page's three sub-questions. We verified everything we could — 650 sampled cases, zero disagreements — but this needs outside eyes before we call it settled.
LIVEFULL WRITE-UPk=2, k=3, and k=4 are fully closed, each independently verified — k=4 cross-checked against a real 1975 paper. k=6's overlapping case is now fully classified too (verified by independent brute force), reduced to four explicit curves for the hard disjoint case — honestly not yet closed, but the exact remaining computation is now specified.
LIVEFULL WRITE-UPErdős #389 itself stays open. Our shift conjecture got refuted by an exact counterexample — then a same-day follow-up proved an exact obstruction formula and certified a 14th minimal witness (n=28), piggybacking on a public exhaustive computation for a different Erdős problem.
LIVEFULL WRITE-UPf(n) counts the biggest strictly-increasing sequence up to n where every contiguous run sums to something different. New result: f(n) ≤ n/2 + √n + O(1) — and we didn't just check the final number, we replayed the entire proof by hand and it never once broke.
LIVEFULL WRITE-UPFour rounds in. Every leading coefficient u is reachable for SOME v, but round 4 found a genuine mod-7 wall: (7,1) specifically is unreachable by this whole construction family — real information, not just a failed search, and we caught and dropped an overreaching claim in the same return.
LIVEFULL WRITE-UPErdős and Graham asked in 1980 whether every set of K-th powers has a tiling complement. We think the answer is yes for every K≥3 — which, with the known negative case at K=2, closes the conjecture in full.
CLOSEDFULL WRITE-UPFound non-log-concave trees, but still unimodal — the target was never met across 11 search runs.
WALLComputational upper bounds for f(4), f(5) verified. General problem open.
PARTIALThe one that beat every model at every budget in our cross-model bench. Best outcome: an honest decline.
WALLVerifier built and checked; no verdict write-up was ever produced.
WALLEarly batch run — no verdict recovered (run JSON only).
WALLStrong partial, verified — not closed by us (Price's claimed proof is unauditable).
PARTIALComplete N=4, k=4 certificate — no integers m, n exist. Full problem stays open.
PARTIAL