A PDF circulating on Hacker News this week claims that Anthropic's Claude Opus 4.6 has produced a complete proof of the Erdős Prime Divisibility Conjecture for Binomial Coefficients — a longstanding open problem in combinatorial number theory first posed by Paul Erdős. The conjecture states that for all integers 1 ≤ i < j ≤ n/2 with n ≥ 2j, there exists a prime p ≥ i dividing the greatest common divisor of the binomial coefficients C(n,i) and C(n,j). If verified, the result would mark one of the most significant autonomous mathematical contributions ever attributed to a large language model.
The document, labeled Revision 4 (Final), pursues the proof across three layers. The algebraic core draws on Kummer's Theorem, Legendre's Formula, and the Sylvester–Schur Theorem, together with two constructions the document introduces as the Prime Power Bridge Lemma and the Cofactor Escape Lemma, which together resolve all but a residual 'fully obstructed' configuration. That configuration is then handled by a Diophantine layer invoking S-unit equations, bounding the problematic cases through what the document calls a Smooth Pair Theorem, with special handling for small values of i. The argument is supported by exhaustive computational verification of over 109 million triples with n ≤ 4400, yielding zero counterexamples.
Reactions in the Hacker News thread were mixed and largely skeptical. Multiple commenters flagged the hosting choice — a file-sharing service rather than arXiv — as an immediate credibility concern. Others engaged the technical content directly, questioning whether the Cofactor Escape Lemma is genuinely novel or whether it recapitulates known results in p-adic valuation theory. At least one commenter identifying a background in analytic number theory expressed cautious interest in the S-unit equations argument but said they hadn't worked through it fully.
Those reservations are hard to dismiss. The PDF has not appeared on arXiv or in any peer-reviewed venue, and no named mathematician has publicly vouched for it. LLMs have a well-documented track record of producing proofs that appear structurally sound but contain subtle errors — often concentrated in precisely the kind of novel lemma constructions doing the heaviest lifting here. The Cofactor Escape Lemma and Smooth Pair Theorem are not established results, which means the proof's validity rests on constructions that cannot be checked against existing literature. Until a researcher with the relevant expertise works through the full argument and puts their name on an assessment, the claim remains unverified.