Simo Ryu @cloneofsimo
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Simo Ryu @cloneofsimo · Aug 30
wow apparently there is now stronger result on large prime gap by GPT 5.6, improving beyond the previous best result of Ford-Green-Konyagin-Maynard-Tao [2014] by factor of log3 X / (log4 X)^2. I recall tao mentioning that it would require genuinely new idea to break his result, and that seems to be the case here.
+ its been verified in lean unconditionally?
erdosproblems.com/forum/thread/4...
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Summary: The paper claims to prove that there are infinitely many n such that
p_{n+1} − p_n > C · (log n · log log n) / (log log log log n)
by proving the interval [x, y] can be covered by a_p mod p, p < x for y ≈ (x log x)/(log log log x).
For primes p ≤ log^100 x, take every number 0 mod p.
For primes log^100 x < p ≤ x/2, pick 0 with probability 1 − β_p, otherwise pick random nonzero residue, each with probability β_p/(p−1). β_p is very close to 0.Note from Claude Sonnet 5
Tweet by Simo Ryu reporting a claimed improvement by GPT 5.6 on large prime gaps beyond the Ford-Green-Konyagin-Maynard-Tao result, with a Lean verification mentioned and an erdosproblems.com forum link; an image excerpt summarises the paper's covering argument.