open problem

The Twin Prime Conjecture

Are there infinitely many primes p with p + 2 also prime? Zhang's 2013 bounded-gaps breakthrough brought it within reach.

4 machine-reviewed papers whose extracted claims name this problem. Matching is literal, over the claim graph, so every listing traces to a sentence in a review.

This is the review hub: papers that name the problem, with the machine's verdict on each. The claims shelf lists the individual claims themselves, formal (Lean) or stated.

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REJECT Sieve methods and the twin prime conjecture
The paper claims a proof of the twin prime conjecture via an induction on an elementary sieve inequality, but the proof is not rigorous.

arxiv:1909.02205 · math.GM · 2019-09
REJECT Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems
A generalized Ramsey number framework is introduced, with polynomial characterizations, and several prime-gap problems are re-expressed as existence of these numbers, largely by construction.

arxiv:2502.04311 · math.CO · 2025-02
REJECT Notes on the Divisibility of Divisor Sums of Catalan Numbers
The paper asserts divisibility properties of Catalan numbers, including 6 | sigma(C_n) for n > 6, but the central proof is invalid and the asymptotics are arithmetically wrong.

arxiv:2502.04619 · math.CO · 2025-02
CONDITIONAL An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime
Every even number up to a trillion is shown by exhaustive computation to be a prime plus a twin prime, and a proof shows the twin member's orientation is forced by the residue of n mod 3.

arxiv:2608.02381 · math.NT · 2026-08