Problems

42 problem folders. Statuses come from the README table; each page renders the folder's PROBLEM, ATTACK, WALKTHROUGH, and RESEARCH files.

• The Erdős–Szekeres number ES(7) (erdos-szekeres-seven)The published record is $33 \leq ES(7) \leq 113$. The classical 32-point witness is independently replayed in Python and C. A compact signotope encoding has a checked small DRAT proof; its full 33-vertex run returned UNKNOWN after 300 seconds, so no bound changed.
• Linear covering codes of radius two (covering)$\ell_2(10,2)\le 50,\ \ell_2(18,3)\le 151,\ \ell_2(21,3)\le 303,\ \ell_2(26,2)\le 13070,\ \ell_2(26,3)\le 817,\ \ell_2(31,4)\le 689,\ \ell_2(36,2)\le 418271,\ \ell_2(38,3)\le 13102,\ \ell_2(41,3)\le 26175,\ \ell_2(44,3)\le52351,\ \ell_2(47,3)\le104703$; $\mathrm{QM}\ 22/24/28$ and $p(H_{18})\le 17,\ p(H_{20})\le 14,\ p(H_{26})\le 19,\ p(H_{28})\le 28$. The redundancy-44 radius-3 construction improves Table 7.2's 52415 by 64, with an exhaustive $2^{26}$ seed-partition check and a columnwise lift identity. A 33-block merge of the same seed gives the redundancy-41 radius-3 bound 26175, improving Table 7.2's 26238 by 63, with the same seed-sweep and lift-identity checks. A 129-block refinement of the same seed (q15) gives the redundancy-47 bound 104703, improving Table 7.2's 104831 by 128. $n=49$ still 7 holes; q9 recovers the 2003 Kaikkonen–Rosendahl 51-set, shows 51 and 50 are not lifts of the $r=8$ record in any of the 174251 quotients, and exhaustively rules out 271127 of the 279034 single-block shrinks of the 50-set with block width $\le 12$. q10 prescribes an automorphism instead of perturbing the 50: order 7 is settled at $r=10$ for every fixed-space dimension, and with orders 11, 17, 31, 73, 127 also excluded, any 49-set has a $\{2,3,5\}$-group of automorphisms. At $r=11$ no invariant set below the record 79 exists for orders 11, 17 or 23. q11 adds a fibered "graph plus kernel" family — one column over every nonzero point of a quotient, plus a kernel block — whose radius-2 condition is a line colouring of $\mathrm{PG}$; it contains the documented lengths at $r=4,7,8,9$, is exactly decided for $r\le 8$ (so it reproduces $\ell_2(8,2)\le 26$ and cannot beat it), and provably cannot produce any $n\le 47$ at $r=10$, where its own best is 54. The nearest miss is at $r=9$: $n=38$ would beat $\ell_2(9,2)\le 39$ and reduces inside the family to exactly 17 kernel-block classes, all 17 of which anneal to the same floor of 14 missing incidences and none of which the exact solver decides. q14 exhausts two local neighborhoods of the 50-set (every two-delete/one-add and three-delete/two-add switch leaves holes); $n=49$ remains open. These constructions are in the joint paper arXiv:2609.16078 with Davydov, Marcugini, and Pambianco.
• Brocard–Ramanujan modular hunt (brocard)Four infinite prime-offset families excluded by Wilson's theorem: $n=p-2$ or $p-3$ for primes $p\equiv3,5\pmod 8$ (with $p>3$ in the second form). Conjecture open.
• Sets with no unique sum mod p (unique-sum)Independently replayed through $p=61$, matching OEIS A398173 at 59 and 61 (the OEIS table continues through 73). Checked 15-sets and completed exclusions of every smaller size give $m(59)=15$ and $m(61)=15$. No published bound improved.
• No-three-in-line at n=75 (three-in-line)Replayed Heule's rct4 142-set at $n=71$: $D(71)=142$. At the first current hole, $n=75$, an audited rct4 portfolio ended without a 150-set.
• Improve the lower bound for the seventh Schur number (schur)No 1697-coloring found.
• Van der Waerden W(2,7) (vdw-w27)Verified Paley/QR coloring of $[3703]$. Does not extend to 3704. No improvement.
• Shannon capacity of C_7, fifth strong power (c7-shannon)Verified Polak–Schrijver 367-set in $C_7^{\boxtimes 5}$. No 368. Missing a letter in any coordinate caps the set at 345. Hamming distance 11 from the published set is empty. No improvement.
• Riemann hypothesis (riemann-hypothesis)Clay Riemann hypothesis; also Hilbert 8(a) and Smale 1. Independent verification of Gomila's candidate gives $0\leq\Lambda\leq\frac{893927}{5000000}=0.1787854<\frac15.$ Two complete finite algorithms, fresh error/tail/barrier computations, and an analytic review are retained in the certificate and note. Prior published upper bound: $0.2$ (Platt–Trudgian). RH remains open.
• Landau 3: Legendre's conjecture (landau-legendre)Landau 3. Assuming RH, primes occur between consecutive $(2+\delta)$-powers for every real $x\geq1$ when $\delta\geq0.22524401991935$, improving the earlier notebook threshold $0.22525$ and the printed $0.2253$. Both Oppermann halves were independently replayed for the last 100,000 square intervals below $2^{64}$. The published finite record remains $n=7.05\cdot10^{13}$, and Legendre's conjecture remains open.
• Landau 4: primes of the form n^2+1 (landau-n2-plus-1)Landau 4. Certified 3954181 primes $n^2+1$ for $n\le 10^8$, matching Wolf $\pi_q(10^{16})$. Infinitude open.
• The second term for Sidon subsets of [N] (sidon-second-term)Independent 11-kernel certificate: $F(N)\le N^{1/2}+0.94301\,N^{1/4}+O(1)$. Beats Hou–Zhao arXiv:2607.01169v2 (0.9435).
• Chowla's cosine problem (chowla-cosine)$K(n)\ge n^{1/7}/18$ for all $n\ge 1$. Does not beat Bedert $n^{1/5-o(1)}$.
• Shrink the 509-vertex five-chromatic unit-distance graph (unit-distance-509)Rebuilt Parts 509, 5-chromatic and vertex-critical. No smaller graph found. Record still 509.
• Chromatic number of the plane (hadwiger-nelson-plane)Published interval $5\le\chi(\mathbb R^2)\le7$. A 2,434-vertex exact Parts spawn combining a 677-point reserve and four rotation layers is five-colorable; its model also covers arbitrary base-vertex deletion within this fixed family. No new lower bound.
• Zeros of integer cosine sums (cosine-zeros)$Z(N)\ge\log\log N/(200\log\log\log N)$ when RHS $\ge 4$. Does not beat Bedert's exponent.
• A smaller constant in the two-squares gap (two-squares-gap)Jameson $a=3$ on $n\le 1.024\cdot 10^{15}$ except $\{3,6,21,91\}$ (m $\le 250$ cert replayed). Green's $1/10$ open.
• Density of the Ulam sequence (ulam-sequence)L=22 word with $C_F<1.442$, beating CS 1.454. Density open.
• A long gap in a dilate modulo a prime (long-gap-dilate)SAT $G(p,\mathrm{round}\sqrt p)$ through $p=73$. No universal $C>2$. Record still Shakan 2.
• Thin cyclic additive bases (thin-cyclic-bases)BEL $\sqrt{8/3}$ family replayed through $q=61$. No thinner liminf. $\sqrt2$ open.
• Frankl's union-closed sets conjecture (union-closed)Certified frequency $0.38305$ on the two-point family {b,1} under Liu Example 4. Analytic first-crossing $0.3830513565\ldots$; a 9,000 by 7,000 Python/C mesh has minimum ratio $1.0000049143$. This improves the prior printed $0.38304$ and Liu's $0.382709$. Frankl's one-half target remains open.
• Exact Cohn–Elkies certificate for planar circle packing (cohn-elkies)Exact $R=3627599/500000=7.255198$, beats printed Table 4 $7.25520$. Not a magic function.
• The five-dimensional kissing number (kissing-5d)All four 40-point codes (D5, L5, Q5, R5) are polar-maximal (polar max $5/4$). Integer Delsarte on the Q5 angles excludes size 44. Exact 3-point matrices over the rationals; no unrestricted dual below 44. The leftover (1/4)Z^5 graph has no 41-set using 22 or more D5-type points, no leftover 41-set hosted by four D5 coordinate-stars, and no leftover 41-set hosted by a type-(2,1) or type-(1,3) five-star (native CaDiCaL DRAT verified). Type-(0,5) leftover-tight SAT and leftover SAT with |U|=19 did not finish (no stored verified DRAT). The 355-point T^5 remainder has no 36-clique (native CaDiCaL DRAT verified). Range still $40\le\tau_5\le 44$.
• Extremal affine copies of {0,1,3} (affine-013)$T(S)\le\lceil n^2/2\rceil$ for affine copies of $\{0,1,3\}$. Beats Aaronson $3/4$. Conjecture $1/3$ open.
• Irreducibility of random 0/1 polynomials (zero-one-polynomials)BSKK $\theta=0.00373556$ (published $0.003736$ over-round). Census $n\le 20$. $p_n\to 1$ open.
• The 1/3–2/3 conjecture for posets (one-third-two-thirds)Gupta v2 is a full order-14 balance census (least above $1/3$ is $37/106$). Width-3 $W_{10}$ at $6/17$ is the width-3 minimum through 14. Broken-rung ladder minima independently replayed through 14 and computed through 22. Three-rail minima through 15 stay above $6/17$. Interval orders through 10 stay at or above $1/3$. Conjecture open.
• Seymour's second-neighborhood conjecture (seymour-second-neighborhood)Explicit n=8 Pisa graphs with irregular missing degrees $3^2 2^6$, $3^4 2^4$, $3^6 2^2$. Seven stored witnesses replayed. Did not re-run the 2.5B census. Conjecture open.
• Two smooth summands for every integer (two-smooth-summands)Incomplete search. $F(131486759)=83$, not a $79$-smooth sum. $G(y)$ through $y=79$ replayed. No exponent below Balog $4/(9\sqrt{e})$. Green #59 open.
• Tuza's triangle packing-covering conjecture (tuza-triangle-packing-covering)8-regular codegree-7 Puleo pair: 1044 cores independently checked (STS(9) on $K_7$). Does not prove Tuza for $\Delta\le 8$.
• Caccetta–Häggkvist for directed triangles (caccetta-haggkvist-k3)Exact CH-triangle at n=18, 21, 24, 26, 27, 29, 30, 32, 33, 35, 36 and every leftover order through n=130 (534 cubes through n=72, 1026 through n=108, 213 through n=114, 36 through n=115, 37 through n=116, 38 through n=117, 37 through n=118, 38 through n=119, 39 through n=120, 38 through n=121, 39 through n=122, 40 through n=123, 39 through n=124, 40 through n=125, 41 through n=126, 40 through n=127, 41 through n=128, 42 through n=129, then 41 through n=130, then 42 through n=131, then 43 through n=132, then 42 through n=133, then 43 through n=134, then 44 through n=135, then 43 through n=136, then 44 through n=137, then 45 through n=138, then 44 through n=139, then 45 through n=140, then 46 through n=141, then 45 through n=142, then 46 through n=143, then 47 through n=144, then 46 through n=145, then 47 through n=146, then 48 through n=147, then 47 through n=148, then 48 through n=149, then 49 through n=150, then 48 through n=151, then 49 through n=152). First remaining hole n=153. $F_4\text{ certificate }c=0.34640\text{ (CKLS 2015 fork), below HKN }0.3465\text{ and the prior stored }0.34645.$ Did not beat 0.3388. Conjecture 1/3 open.
• Existence of a projective plane of order 12 (projective-plane-order-twelve)Two involution 2-MOLS replayed (intercalates $108+108$ vs $90+78$). $t=3$ timeout. Published Aut still $\lvert G\rvert\in\{1,2,3\}$. Plane open.
• Determine R(5,5) (ramsey-r55)McKay 328+328 (5,5,42) graphs replayed; circulant 42/43 empty. A hypothetical 43-vertex example cannot have an automorphism of order 7, nor of cycle type 5^6 or 5^7, nor of type 5^4 with a fixed vertex meeting 0 or 4 of the 5-cycles, nor of type 5^5 with a fixed vertex meeting 2 or 3 of the 5-cycles. Leftover automorphism primes are 2, 3, and 5. Interval still $43\le R(5,5)\le 46.$
• Push graph reconstruction beyond 13 vertices (graph-reconstruction-next-order)Full independent degseq census: all 8,571,837 n=14 graphs are $4^{11}6^{3}$; 17,143 labelg samples matched. Uniqueness (hence reconstructibility) not independently re-sorted. McKay n=13 unchanged.
• Lonely runner for 14 runners (lonely-runner-fourteen)Every 13-tuple of coprime speeds with $u_i \equiv i \pmod{p}$ has the lonely runner property, for each of the ten primes 191 through 239, checked exhaustively in about four minutes apiece. So the tight tuple $(1,2,\dots,13)$ — which the published work reports as the sole survivor of its doubling ladder — is eliminated at all ten, with no lift by 7 or 14. The check fails for primes up to 41, so the modulus is doing real work. Sungkawichai–Trakulthongchai (arXiv:2604.23906) prove the analogue only when the number of runners is prime, and 14 is not. Cross-checked against brute force at 6 and 8 runners. Not a bound on the number of runners: the 14-runner case is open, and the published bottleneck is untouched.
• Elliptic Fekete / log-energy points on S^2 (fekete-s2)Replayed Ridgway–Cheviakov 2018 Table 3 for $N=2$–$65$. No improvement. Smale 7 open.
• Smale 16 — the plane Jacobian conjecture (smale-jacobian-n2)Smale 16 / plane Jacobian conjecture. Guccione–Guccione–Horruitiner–Valqui left only $(72,108)$, up to transposition, below maximum degree 125. A pinned exact unit-ideal certificate replay, with an independent Python/Rust check of the published polygon bridge, excludes that pair. Thus every hypothetical plane counterexample has $\max\{\deg P,\deg Q\}\ge125.$ The conjecture remains open; the 2026 three-variable announcement is not treated as community consensus.
• Smale 4 — the Shub–Smale τ-conjecture (smale-tau)Smale 4 / Shub–Smale $\tau$-conjecture. The polynomial form has no finite record to move, so the campaign takes the integer question that Smale states in the same problem, whether $\tau(k!)\le(\log k)^c$. An exhaustive 12-step search with exact arithmetic gives $\tau(29\#)=\tau(31\#)=12,$ where Markström (arXiv:1306.3091) had at most 13 and 15; it replays the OEIS A217032 record through 19! and extends the A173419 table to $n\le 10266$. The 13-step decision for 20!, 21!, 22! is residue (incomplete). Exact enumeration over integer polynomials gives the largest number of distinct integer zeros at cost $k\le 6$ as 1, 1, 2, 3, 3, 4, 5. Conjecture open.
• Hilbert 16(a) for degree 8 — real schemes of plane octics (hilbert16-degree-8)Number of nonempty degree-eight T-curve schemes ≥ 2,415. Eight schemes lie outside the 2,407 baseline formed by the 2,367 certificates of arXiv:2602.06888v4, the notebook’s seventeen prior additions, ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩, nine later additions, and thirteen more: ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩, ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩, ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩, ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩. Exact integer liftings and independent Python/Rust topology checks pass; claim and certificates. Neither open maximal deep nest is decided. Hilbert 16(a) degree 8 remains open.
• Hilbert 16(b) — the Hilbert number H(n) (hilbert16-limit-cycles)Hilbert 16(b): the Hilbert number H(n), maximum isolated periodic orbits of a planar polynomial vector field of degree n. Published lower bounds still stand: H(2) at least 4 (Shi; Chen–Wang), H(3) at least 13 (Li–Liu–Yang), H(4) at least 28 (Prohens–Torregrosa), and the Han–Li / Chebyshev lifts. A quadratic with 5 cycles, a cubic with 14, a homogeneous nest, a Liénard beat of B(n), iterated squaring past the quadratic ceiling, a Harnack-recurrence table beat, a 29th cycle on the Prohens–Torregrosa quartic, a two-well 14, a four-zero quasi-homogeneous Melnikov, a 9-sheet holomorphic cube, a beat of Z(2,n), a constructive +1 with two cycles, a cubic with an invariant line and three cycles, five Abelian zeros, a four-fold or T2 beat of H(7) at least 74, a cubic Kolmogorov with 7 cycles, an L2 extra nest, and a degree-5 radial product with two cycles were not constructed. Replayed: Shi’s order-3 jet, van der Pol uniqueness, the Chebyshev pullback identity and its 2026 table arithmetic, uniqueness for one radial cubic, the first Lyapunov quantity of a quadratic focus, homogeneous scaling, the odd-cubic Liénard energy identity, iterated-squaring sheet counts, the Harnack-recurrence arithmetic, the explicit degree-4 Darboux seed with three centers, the two-well energy and figure-eight integral, vanishing of L1 at those three centers, first-order cyclicity at most 1 for one cubic perturbation of the quasi-homogeneous center, the holomorphic-cube sheet count, the radial family attaining Z(2,n), a degree-4 line product with one circle, a named cubic with an invariant line and no cycles, an explicit Christopher–Lloyd degree-7 four-oval field, an explicit T2 degree-7 four-oval field, a cubic Kolmogorov Dulac in the first quadrant, the second Poincaré quantity at the two-well, and the degree-5 radial product with one circle. No published H(n) moved. Different problem from the degree-8 folder above.
• Lieb–Thirring conjecture (Simon 2000, #15) (simon-lieb-thirring)Later record is Carvalho Corso–Ried (arXiv:2403.04347v2) $L_{1,1,1}/L_{1,1,1}^{\mathrm{cl}}\le 1.44655\quad(M_3=0.371185695).$ Independently the Clausen form of that value is at most 1.4465531. Did not beat 1.44655. Weidl interpolation, remainder absorption, a Neumann covering, and a 1D test-potential search also did not beat 1.44655. The 2018 FHJN trial pair was 1.456; a certified pair on this notebook still gives 1.45576 and does not beat CCR. Conjecture open.
• Simon 2000 #9: bounded excess charge (simon-ionization-excess)Simon 2000 #9. Replayed Lieb $2Z+1$, Nam $1.22$, and Hundertmark–Pattakos–Schulz. Remainders $2.953$, $3.892$, $3.9781$ from the $Z\ge 4$ ratio. Leading coefficient moved from printed $1.1185$ through $1.1118$, $1.1057$, $1.1035$, $1.1026$, $1.1021$, $1.1020$, $1.1017$, $1.1013$, $1.1010$, and $1.1006$ to $1.1005$. Two independent rational matrix checks prove $N_c(Z)<1.1005Z+3.933Z^{1/3}$ for $Z\ge4$; claim and replay. Hydrogen N0(1)=2 replayed. Bounded excess open.
• Continuity of the integrated density of states (simon-ids-continuity)Simon 2000 #14. Bourgain–Klein proves log-Hölder continuity for arbitrary bounded continuum potentials in dimensions 1–3 and for discrete operators in every dimension. An exact Python/Rust replay locates their continuum frontier at $d=4$, where the modulus exponent $(4-d)/8$ becomes zero. Frank–Ivanisvili's 2026 real bounded Landis examples make the underlying $4/3$ power sharp for this route. Adding a free Euclidean direction gives an absolutely continuous DOS by the standard separable convolution identity. General bounded ergodic continuum potentials in dimensions at least 4 remain open.