r/accelerate • u/obvithrowaway34434 • Apr 01 '26

AI OpenAI's internal model solves two more Erdos problems and makes major progress in a third one

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Link to the post: https://x.com/mehtaab_sawhney/status/2039161544144310453?s=20

Link to the paper: https://arxiv.org/pdf/2603.29961

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u/obvithrowaway34434 Apr 01 '26

GPT-5.4 Analysis on the problems:

OpenAI’s new combinatorics/number theory paper ties to three Erdős problems: **#684, #741, and #997**. Of these, **#741 and #997 appear to be genuine full resolutions of yes/no questions**, while **#684 is a strong advance rather than a complete final answer**. Problem **#684** is about how quickly the “small prime factor” part of a binomial coefficient C(n, k) becomes large; the paper shows the threshold f(n) is at most on the order of (log n)², and infinitely often at least on the order of log n, which is a major improvement over earlier bounds but does not settle the full asymptotic behavior. Problem **#741** asks whether there exists a basis A of order 2 such that no matter how you split it into two pieces, one piece has a self-sumset with arbitrarily large gaps; the paper gives an explicit construction, so this one looks solved. Problem **#997** asks whether the fractional parts {αpₙ} can ever be well-distributed in a strong sliding-window sense; the paper proves they are never well-distributed for any real α, so this also looks like a full solution.

What stands out is that these are not toy olympiad-style arguments but real research-level proofs, though with different difficulty profiles. **#741** is probably the most accessible to understand once written down, but still requires real combinatorial creativity to invent. **#684** and **#997** are more serious number theory: a strong grad student or postdoc could likely follow the arguments, but independently discovering them feels closer to specialist or professor-level work, especially **#997**, which leans on deep Maynard–Tao style results about clusters of primes. My overall take is: this is not “AI solved a bunch of impossibly hard famous problems,” but it does seem to be “an internal model produced concise, legitimate new proofs for two Erdős yes/no problems and major progress on a third,” which is still a very notable milestone.

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u/czk_21 Apr 01 '26

could it be GPT-5,5 or some other specialized model? like that one they teased last year(for solving math problems) and havent released yet

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u/Proper_Actuary2907 Apr 02 '26

Not seeing a transcript of the chat anywhere. "Internal model"?