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Public Falsification Record

Every claim killed by the three-gate verification pipeline: sealed-sandbox reproduction failure (Gate 2), adversarial red-team attack (Gate 3), or mathematical counterexample (Gate 1). Total: 94.  · JSON

33 Gate 2/3 pipeline kills  ·  61 math counterexample kills

Gate 2 / Gate 3 Pipeline Falsifications (33)

Claims VERIFIED at Gate 2 (sealed-sandbox repro) and subsequently falsified by the Gate 3 adversarial red-team (three independent LLM attackers, inverted scoring). A claim SURVIVES only if all three attackers fail to find a fatal flaw (avg attack score < 3.5; no individual score ≥ 5.0).

Task ID Gate Claim type Goal / Claim Avg attack Killed (UTC)
0595bd4f-0470-40… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 6.5/10 · REPLICATION ATTACKER: 7.5/10
 7.5/10 2026-06-11 01:23
50a4f525-1f3f-43… Gate 3 formula_repro What is the performance degradation of Unified-IO 2 on the VQA-v2 dataset when audio modalities are introduced as distractors versus text-on…
COUNTEREXAMPLE HUNTER: 7.5/10 · CITATION AUDITOR: 5.0/10 · REPLICATION ATTACKER: 7.5/10
 6.7/10 2026-06-11 01:22
f82ac2f4-1a92-4e… Gate 3 formula_repro How does GRACE's quantization-aware training scale with model size, and how does it affect performance on the MME and MM1K benchmarks when a…
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 7.2/10 · REPLICATION ATTACKER: 2.0/10
 5.9/10 2026-06-11 01:22
cc2d0e37-a950-4a… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 3.5/10
 7.0/10 2026-06-10 19:35
673590a7-25e9-41… Gate 3 formula_repro How does Qwen3's performance on GPQA Diamond compare to other frontier models when evaluated under chain-of-thought prompting versus standar…
COUNTEREXAMPLE HUNTER: 0.0/10 · CITATION AUDITOR: 9.2/10 · REPLICATION ATTACKER: 9.5/10
 6.2/10 2026-06-10 19:35
b472d355-87a8-45… Gate 3 formula_repro How do language models compare to human experts on professional knowledge and science benchmarks v19
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 7.5/10 · REPLICATION ATTACKER: 9.5/10
 8.7/10 2026-06-10 19:34
b5058ffc-3f4d-46… Gate 3 formula_repro What is the impact of million-token context windows on multimodal reasoning accuracy in Gemini 1.5 Pro versus prior versions?
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 9.5/10
 8.8/10 2026-06-10 19:34
09acaf30-ab81-49… Gate 3 formula_repro To what extent does chain-of-thought prompting mitigate performance degradation in long-horizon reasoning tasks for LLMs evaluated on the Bi…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.3/10 2026-06-10 19:34
031cd03f-2fbe-4d… Gate 3 formula_repro What are the benchmark performance scores of GLM-4.5-Air on reasoning mathematics coding and language understanding tasks
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 0.0/10 · REPLICATION ATTACKER: 8.5/10
 5.8/10 2026-06-10 19:34
99e0cc2f-ae34-40… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.2/10 · REPLICATION ATTACKER: 8.5/10
 8.9/10 2026-06-10 16:52
73ec2b2b-e67b-47… Gate 3 formula_repro What is the cross-domain generalization capability of OpenPangu-7B-MLA on empathetic speech understanding tasks when evaluated on MMSU and o…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.3/10 2026-06-10 16:51
dd13f070-1013-42… Gate 3 formula_repro How does the performance of self-supervised foundation models on tabular data classification compare to standard normalization techniques wh…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 9.5/10
 9.0/10 2026-06-10 16:51
2483aaac-7f84-4c… Gate 3 formula_repro To what extent does fine-tuning on adversarial multi-hop QA examples improve the robustness of RAG systems against distractor contexts compa…
COUNTEREXAMPLE HUNTER: 9.5/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.5/10 2026-06-10 16:51
3cbe8120-1209-45… Gate 3 formula_repro How does fine-tuning on AdvRACE affect the cross-lingual robustness of MRC models when evaluated on adversarial perturbations in non-English…
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.2/10 2026-06-10 16:51
4a909146-446f-4d… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 7.5/10 · REPLICATION ATTACKER: 2.5/10
 6.3/10 2026-06-10 10:48
426ccfd7-06e6-40… Gate 3 formula_repro How does the integration of non-lexical vocal cues in multimodal language models like OpenPangu-7B-MLA affect downstream task performance on…
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 3.5/10 · REPLICATION ATTACKER: 7.5/10
 6.5/10 2026-06-10 10:47
294a5d5b-f300-40… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 0.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 8.5/10
 5.7/10 2026-06-10 08:45
18e28019-37fb-4c… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 0.0/10 · CITATION AUDITOR: 7.5/10 · REPLICATION ATTACKER: 9.2/10
 5.6/10 2026-06-10 08:45
a80b4a8e-8700-4c… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 9.2/10
 8.9/10 2026-06-10 08:44
e26d33b4-a5b3-48… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 6.5/10 · REPLICATION ATTACKER: 3.0/10
 6.2/10 2026-06-10 08:44
30bd9c9a-90c8-4e… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 9.2/10
 8.9/10 2026-06-10 08:43
3b783c5d-ec77-4e… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 0.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 9.2/10
 5.9/10 2026-06-10 08:42
388b9655-1a81-4e… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 0.0/10
 5.8/10 2026-06-10 08:42
42863d1d-2f6a-41… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 8.5/10
 8.7/10 2026-06-10 08:42
0e47786d-3f42-43… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 7.5/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 8.5/10
 8.2/10 2026-06-10 08:41
3904006d-6cfc-42… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 8.5/10
 8.7/10 2026-06-10 08:41
845a22c0-61ad-4e… Gate 3 arithmetic_repro -
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 8.5/10 · REPLICATION ATTACKER: 8.5/10
 8.7/10 2026-06-10 08:41
42a5d013-2da3-4d… Gate 3 unknown -
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.2/10 2026-06-10 08:36
8520660f-c1c4-4c… Gate 3 unknown -
COUNTEREXAMPLE HUNTER: 0.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 6.3/10 2026-06-10 08:36
fa1dffe8-f9a9-4f… Gate 3 formula_repro How does the F1-score of diffusion-based tabular generative models compare to CTGAN when augmenting data for training LLMs on imbalanced tex…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.3/10 2026-06-10 08:35
ee851b65-000d-44… Gate 3 formula_repro What is the impact of varying the pretraining dataset size and diversity on the cross-domain generalization capabilities of tabular foundati…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.3/10 2026-06-10 08:35
11c29061-cf3e-4b… Gate 3 formula_repro Does scaling the size of domain-specific training data for RAG models improve alignment with human evaluators when measured by RAGalyst's me…
COUNTEREXAMPLE HUNTER: 8.5/10 · CITATION AUDITOR: 7.5/10 · REPLICATION ATTACKER: 7.5/10
 7.8/10 2026-06-10 08:35
9f6b0926-918c-40… Gate 3 formula_repro How does the scaling of unlabeled video-audio pretraining data affect the few-shot adaptation accuracy of latent action models on the RoboBe…
COUNTEREXAMPLE HUNTER: 9.0/10 · CITATION AUDITOR: 9.5/10 · REPLICATION ATTACKER: 9.5/10
 9.3/10 2026-06-10 08:35

Math Counterexample Kills (61 total)

Conjectures generated by the autonomous math research pipeline and killed at Gate 1 when a numerical counterexample was found. These never reach the Lean 4 proof stage.

Conjecture ID Problem Statement (falsified) Killed (UTC)
126dd256f8354b53… Sum of Odd Numbers Identity The sum of the first 100 odd positive integers equals 10,000. 2026-06-10 22:08
3f7781c61e534635… Square Minus Square Factoring For every natural number n less than 100, the square of n modulo 2 is either 0 or 1. 2026-06-10 18:05
8f940771d9454664… Square Minus Square Factoring For every natural number n from 0 to 99, the square of n modulo 2 is either 0 or 1. 2026-06-10 18:05
b33154755b914337… Square Minus Square Factoring For every natural number n less than 100, the square of n is congruent to either 0 or 1 modulo 2. 2026-06-10 18:05
208383baf95c4350… Sum of Odd Numbers Identity The sum of the first 100 odd positive integers equals 100 squared. 2026-06-10 09:18
5e66db8a98eb4f41… Sum of Odd Numbers Identity The sum of the first 150 odd positive integers equals 150 squared. 2026-06-10 09:18
7f248d0d38c24d74… Goldbach conjecture — computational extension For every even integer n >= 100, there exists a Goldbach partition n = p + q (with p <= q) such that the prime p satisfies p > n/2 - sqrt(n) * (ln ln n)^2, AND p is a quadratic residue modulo the smallest prime factor of… 2026-06-10 07:28
c37b8d2825f34f46… Primes of form n^2+1 — density and distribution Let S(x) be the set of integers n in [1, x] such that n^2 + 1 is prime. For any two consecutive elements a, b in S(x) (with a < b), the gap g = b - a satisfies g < 2.5 * sqrt(a) * ln(a) for all x >= 1000. This conjecture… 2026-06-10 07:25
007bb20f4be4478b… Ramsey multiplicity K_4 — minimum number of monoch In any 2-coloring of the edges of K_18 that achieves the global minimum number of monochromatic K_4 subgraphs, the resulting color classes (graphs) must both be isomorphic to the Turán graph T(18, 3). Consequently, the m… 2026-06-10 01:47
f600ae4401434fed… Fibonacci primes — density conjecture For every Fibonacci prime F_p with prime index p > 3, the quantity (F_p - 1) / p is never an integer. In other words, no Fibonacci prime (beyond F_3=2 and F_4=3, though 4 is not prime index, specifically checking p=5, 7,… 2026-06-09 13:24
e0243ef726e64075… Collatz conjecture — structural pattern search For any integer n > 1, let S(n) be the set of distinct values visited in the Collatz trajectory of n before reaching 1. Let M(n) be the maximum element in S(n). The conjecture states that the ratio of the count of odd nu… 2026-06-09 00:53
45b6484654eb41d2… Goldbach conjecture — computational extension For every even integer n > 6, there exists a Goldbach partition n = p + q (with p <= q) such that the smaller prime p satisfies p > sqrt(n) and the product p*q is congruent to 1 modulo 24. 2026-06-09 00:52
b171ab227ec34a92… Primes of form n^2+1 — density and distribution Let P be the set of primes of the form n^2+1. For any x >= 10, let S(x) be the sum of the reciprocals of the square roots of the generators n for all such primes p = n^2+1 <= x. The conjecture states that S(x) is strictl… 2026-06-08 20:48
afae0570267f4f10… Twin prime density — Hardy-Littlewood conjecture v For all integers x >= 10,000, the relative error between the actual count of twin prime pairs up to x and the Hardy-Littlewood prediction (2*C2*x/ln(x)^2) is strictly bounded by the function 1.8 / ln(x). Specifically, |p… 2026-06-08 07:37
b77b197f38ef42b4… Ramsey R(4,6) — computational bounds In any 2-coloring of the edges of K_35 that avoids a red K_4 and a blue K_6 (if such a coloring exists), the maximum degree of any vertex in the red subgraph must be strictly less than 12. That is, Δ(Red) ≤ 11. 2026-06-08 07:37
b9da4d4e215345c4… Fibonacci primes — density conjecture For all integers n > 4, if the nth Fibonacci number F_n is prime, then n is either prime itself or n=4. Furthermore, for every prime index p > 3 such that F_p is composite, F_p possesses at least one prime factor q such … 2026-06-07 23:17
24d0c43564104d67… Goldbach conjecture — extend computational verific For every even integer n > 10,000, there exists a Goldbach partition n = p + q (where p and q are primes) such that both p and q are 'isolated' within a window of size W(n) = floor(0.8 * ln(n) * ln(ln(n))). Specifically,… 2026-06-07 15:00
ab2f6f8062e94761… OEIS A001065 — perfect number conjecture For every even perfect number n > 6, the sum of the squares of its proper divisors is strictly congruent to 1 modulo the square of its associated Mersenne prime exponent. Specifically, if n = 2^(p-1)(2^p - 1) where p and… 2026-06-07 14:59
bc6085714d2046c4… Primes of form n^2+1 — density and distribution For the sequence of primes of the form p = n^2 + 1, let n_k be the k-th positive integer such that n_k^2 + 1 is prime. The conjecture states that for all k >= 2, the gap between consecutive bases n_k and n_{k-1} satisfie… 2026-06-07 06:31
9f7d6c51e37b4088… Twin prime density — Hardy-Littlewood conjecture v For all x >= 1000, the actual count of twin prime pairs up to x strictly exceeds the standard Hardy-Littlewood prediction (2*C2*x/ln(x)^2) but remains bounded above by the prediction augmented with a specific second-orde… 2026-06-06 18:04
845aaff7aef64a01… Fibonacci primes — density conjecture For every integer n >= 3, if the Fibonacci number F_n is prime, then n must be a prime number, AND the index n satisfies the property that 2n+1 is either a prime number or a semiprime (product of exactly two primes, not … 2026-06-06 09:39
1028c83002d64c6f… Catalan's conjecture (Mihailescu) — Lean4 formal p For any integer n > 1, if n is a perfect power (n = x^a with x, a > 1) and the next consecutive perfect power m (m = y^b with y, b > 1, m > n) satisfies m - n = 1, then n must be 8. Furthermore, for any perfect power n >… 2026-06-06 05:32
59707dec0c84466f… OEIS A001065 — perfect number conjecture For every even perfect number n > 6, the sum of the binary digits of (n/2) is strictly less than the number of distinct prime factors of (n-1). 2026-06-06 01:23
e1ea6af8e40f4d3a… Collatz conjecture — structural pattern search For any integer n > 1, let S(n) be the set of odd numbers encountered in the Collatz trajectory of n before reaching 1. Let m = min(S(n)). Then the total stopping time (number of steps to reach 1) is strictly less than m… 2026-06-05 21:15
52c749523bfe490f… Primes of form n^2+1 — density conjecture For any integer n >= 2, let S_n be the set of primes of the form k^2+1 where k <= n. Let M_n be the maximum gap between consecutive elements in the sorted sequence S_n (defining the first gap as p_1 - 2). Then, M_n is st… 2026-06-05 16:24
b01e6c25195044a2… Primes of form n^2+1 — density conjecture For every integer n >= 1, the count of primes of the form k^2 + 1 with k <= n (denoted P(n)) satisfies the inequality P(n) >= floor(1.2 * sqrt(n) / ln(n)). Furthermore, for any n >= 100 where P(n) > 0, the gap between co… 2026-06-05 16:22
20a77f3e6ff34241… Fibonacci primes — density conjecture For every Fibonacci prime F_p with index p > 5, the integer part of the square root of the index p, denoted as floor(sqrt(p)), is always a prime number. 2026-06-04 23:37
d177586b3b3d4762… Primes of form n^2+1 — density and distribution Let P_N be the set of primes of the form n^2+1 for 1 <= n <= N. Let A_N be the count of such primes where the generator n is itself a prime number. The conjecture states that for all N >= 1000, the ratio of the density o… 2026-06-04 10:33
498269cc76514396… Twin prime density — Hardy-Littlewood conjecture v The normalized error term of the twin prime count, defined as E(x) = (pi_2(x) * ln(x)^2) / (2 * C2 * x) - 1, exhibits a persistent negative bias for all x in the range [10^4, 10^8]. Specifically, the conjecture states th… 2026-06-03 22:07
b81ab2bf8a3742e6… OEIS A001065 — perfect number conjecture For any even perfect number n > 6, let p be the unique Mersenne prime such that n = 2^(p-1)*(2^p - 1). The sum of the divisors of the exponent (p-1), denoted sigma(p-1), is strictly less than the square root of the Merse… 2026-06-03 07:23
af2e36aa3e2c473a… Primes of form n^2+1 — density and distribution For the sequence of primes of the form n^2+1, let p_k be the k-th such prime. The conjecture states that for all k >= 2, the gap between consecutive primes p_k and p_{k-1} satisfies: p_k - p_{k-1} < 2 * sqrt(p_k) * (ln(p… 2026-06-03 02:25
3c752084d9a043ca… Primes of form n^2+1 — density conjecture For every integer n >= 2, let S_n be the set of primes of the form k^2+1 with k <= n. Let M_n be the maximum gap between consecutive elements in S_n (with the first element treated as having a 'gap' from 0). Then M_n < 4… 2026-06-02 22:15
7c85eafa9f3a4c9b… Twin prime conjecture — density analysis For every integer N >= 100, let T(N) be the count of twin prime pairs (p, p+2) with p <= N, and let S(N) be the sum of the reciprocals of the smaller primes in these pairs (i.e., sum(1/p) for all such p). The conjecture … 2026-06-02 11:16
17b23802a7a14aa9… Cap set problem — F_3^n maximum Conjecture: For n=6, the maximum size of a cap set in F_3^6 is exactly 112, and this maximum is uniquely achieved (up to affine equivalence) by the set of vectors with weight congruent to 1 modulo 3 in the specific coord… 2026-06-02 04:47
1bc7acdee264452e… Catalan's conjecture (Mihailescu) — Lean4 formal p For any integer n > 1, if n is a perfect power (n = x^a with x, a > 1), then the interval (n, n + n^(2/3)] contains no other perfect powers, except for the specific case where n = 8 (2^3), in which case the interval (8, … 2026-06-02 04:44
3f02ace5c31a4891… Goldbach conjecture — computational extension For every even integer n > 100, there exists a Goldbach partition n = p + q (with p <= q) such that the prime p lies within the interval [n/2 - sqrt(n), n/2]. Furthermore, the smallest such prime p satisfies the stronger… 2026-06-01 21:26
2d0c51fde717499c… Primes of form n^2+1 — density and distribution For all integers n >= 2, the gap between consecutive primes of the form k^2+1 is strictly less than 4 * sqrt(p_m) * ln(p_m), where p_m is the smaller prime in the pair. Furthermore, the ratio of the actual gap to this bo… 2026-06-01 17:21
2229477e7b1a459e… Primes of form n^2+1 — density and distribution For the sequence of primes of the form n^2+1, let p_k = n_k^2+1 be the k-th such prime. The conjecture states that for all k >= 2, the gap between consecutive bases n_k and n_{k-1} satisfies: n_k - n_{k-1} < 2 * sqrt(n_{… 2026-06-01 17:18
3032040e036b4ec0… Primes of form n^2+1 — density conjecture For every integer n >= 100, the number of primes of the form k^2+1 with k <= n is strictly greater than the number of primes of the form k^2+1 with k <= n/2 multiplied by the factor (1.3 * sqrt(n) / ln(n)). This conjectu… 2026-06-01 13:41
e2f7b4d3db414cd8… Twin prime conjecture — density analysis For every integer N >= 100, let T(N) be the count of twin prime pairs (p, p+2) with p <= N. The ratio of the actual count T(N) to the Hardy-Littlewood estimate E(N) = 2 * C_2 * N / (ln N)^2 (where C_2 is the twin prime c… 2026-06-01 01:14
6d026f496bb045ec… Fibonacci primes — density conjecture For every integer n > 4, if the n-th Fibonacci number F_n is prime, then n must be a prime number p such that 5 is a quadratic non-residue modulo p (i.e., the Legendre symbol (5/p) = -1). This implies that all Fibonacci … 2026-05-31 21:04
1b3e58ad75384c33… Fibonacci primes — density conjecture For all integers n > 6, if the nth Fibonacci number F_n is prime, then n must be a prime number that can be expressed as the sum of two squares (i.e., n is 2, or n is a prime congruent to 1 modulo 4). This implies that n… 2026-05-31 21:03
a29125de7d584756… Catalan's conjecture (Mihailescu) — Lean4 formal p For any integer n > 1 that is not 8, if n is a perfect power (n = x^a with x>1, a>1), then the smallest perfect power m > n (where m = y^b with y>1, b>1) satisfies the gap inequality m - n > n^0.55. The only exception to… 2026-05-31 16:59
47c3e171d2ff4226… OEIS A001065 — perfect number conjecture For any even perfect number n > 6, let m = n/6. The sum of the proper divisors of m (denoted s(m)) is strictly greater than the square of the number of distinct prime factors of m (denoted omega(m)^2). 2026-05-31 12:56
9c24c2404c5b4ea2… Goldbach conjecture — computational extension The sum of two primes representing an even number n > 2 has its maximal prime difference bounded by n^(0.51), where the exponent 0.51 is strictly between 0.5 and 1. This refines the trivial bound of n-3 by showing the di… 2026-05-31 08:39
5a54ab3135de44a4… Primes of form n^2+1 — density and distribution For integers n >= 2, let P(n) be the set of primes of the form k^2+1 less than or equal to n. Let G(n) be the maximum gap between consecutive elements in P(n) (with the first gap defined as p_1 - 2). The conjecture state… 2026-05-31 05:55
815317887cf646b1… Primes of form n^2+1 — density and distribution For any integer N >= 100, let S_N be the set of primes p <= N such that p = k^2 + 1 for some integer k. Let M_N be the maximum gap between consecutive elements in the sorted sequence S_N (with the first gap defined as th… 2026-05-31 05:55
3e68141113f44ca9… Primes of form n^2+1 — density conjecture For every integer n >= 1, the number of primes of the form k^2 + 1 with k <= n is strictly less than 2 * sqrt(n). Furthermore, the ratio of this count to sqrt(n) never exceeds 1.8 for any n >= 100. 2026-05-31 01:48
21e6bfada240446d… Primes of form n^2+1 — density conjecture For every integer n >= 2, the number of primes of the form k^2 + 1 with k <= n is strictly greater than the number of integers k <= n such that k^2 + 1 is a product of exactly two distinct primes, both of which are congr… 2026-05-31 01:48
3f50b59f69c24ee3… Twin prime density — Hardy-Littlewood conjecture v For all integers x >= 10,000, the cumulative count of twin prime pairs pi_2(x) strictly exceeds the first-order Hardy-Littlewood approximation L_1(x) = 2*C_2 * x / (ln x)^2, but remains bounded above by a second-order co… 2026-05-30 17:37
84e7e3b311544ebb… Cap set problem F_3^6 — verify maximum size = 112 The maximum cap set size in F_3^6 is exactly 112, and this bound is achieved only by the canonical construction S_3^6 ⊂ F_3^6 2026-05-30 04:42
809a9ab0175448e8… Fibonacci primes — density conjecture For all integers n >= 3, if the nth Fibonacci number F_n is prime, then the index n must be a prime number p such that p is not a Wieferich prime base 2 (i.e., 2^(p-1) is not congruent to 1 modulo p^2). Furthermore, for … 2026-05-30 04:30
fe5aa22c047044f1… Cap set problem — F_3^n maximum The maximum size of a cap set in F_3^n for n ≤ 8 is bounded above by ⌊2.2^n⌋, and for n = 6, 7, 8 the values are exactly 124, 353, and 994 respectively 2026-05-30 01:15
23f6590eb1bc458c… Cap set problem — F_3^n maximum The maximum size of a cap set in F_3^n for n=6 is exactly 112, and this value is achieved by a specific construction based on the Edel's bound. 2026-05-30 01:13
028910cf4158418c… Primes of form n^2+1 — density conjecture The count of primes of the form n^2+1 up to a given bound is asymptotically equal to 2*C*Li(x) where C is a constant approximately 0.685 and Li(x) is the logarithmic integral, with the constant C being related to the pro… 2026-05-29 19:18
aac0f88db762449e… Ramsey multiplicity K_4 — minimum number of monoch In any 2-coloring of K_18, the minimum number of monochromatic K_4 is exactly 18, and this minimum is achieved only by colorings where the graph of one color forms a specific structured graph related to the Turán graph T… 2026-05-29 16:36
d45079e02da54eb1… Ramsey multiplicity K_4 — minimum number of monoch In any 2-edge-colored complete graph K_n, the number of monochromatic K_4 subgraphs is minimized by the balanced random coloring. Specifically, for n=18, we conjecture that among all possible 2-colorings of K_18, the min… 2026-05-29 16:29
c08b1bbcbcce45cd… Twin prime density — Hardy-Littlewood conjecture v The number of twin prime pairs up to x follows a cubic correction term in the Hardy-Littlewood approximation, such that the count is better approximated by 2*C2*x/ln(x)^2 + k*x/(ln(x)^3) for some constant k, with the cor… 2026-05-29 11:08
948db96904504e71… Zarankiewicz z(n,n;3,3) — improve upper bound For n≥3, the maximum number of ones in an n×n matrix avoiding a 3×3 all-ones submatrix is strictly less than n^2, and the difference n^2 - z(n,n;3,3) grows at least linearly with n. 2026-05-29 08:52
39b2ddadc40c4457… Van der Waerden w(2;4,4) — verify equals 35 w(2;4,4) = 35: every 2-coloring of {1,...,35} contains a monochromatic arithmetic progression of length 4. Verify computationally by showing all 2-colorings of {1,...,34} avoid monochromatic AP-4 (proving w > 34), and {1… 2026-05-29 07:25
e14fe4d682a542df… Catalan's conjecture (Mihailescu) — Lean4 formal p For any integer n > 1, if there exists a perfect power y^b (with y>1, b>1) such that n < y^b < n + n^(0.6), then n cannot be a perfect power x^a (with x>1, a>1) unless n=8. In other words, the gap between consecutive per… 2026-05-29 06:33