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erdos problems

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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...

[embedded image]
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.

prime gapsgpt 5.6erdos problemsleanai mathematics

@Clambr0

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Costa @Clambr0 [Follow]

hello Erdős asked if Σ n!/(2n)! is irrational

It is. Five-line proof. Extends to the whole family Σ n!/((a+1)n+b)! for a≥1, 0≤b≤a. Lean verified.

My AI then went crazy and claimed the whole affine family a≥1, b≥1-a is transcendental using techniques beyond my understanding:

[link card]
clambro/erdos-270-transcendence
Transcendence in the Affine Case of Erdős Problem 270
1 contributor · 0 issues · 0 stars · 0 forks
GitHub - clambro/erdos-270-transcendence: Transcendenc...
From github.com

2:26 PM · Aug 22, 2026 · 71.2K Views

10 replies · 30 reposts · 465 likes · 193 bookmarks

Relevant ▾        View quotes >

[next post in thread]
Costa @Clambr0 · Aug 22

[embedded image of typeset math, bottom partly covered by phone nav bar]

Let

    C = Σ_{n=1}^∞ n!/(2n)! = Σ_{n=1}^∞ 1/D_n,    D_n := (2n)!/n!.

Since

    D_{n+1}/D_n = 4n + 2 ∈ ℤ,

we have D_n | D_{n+1}.

Let P_N = Σ_{n≤N} 1/D_n and R_N = C − P_N. Then D_N P_N ∈ ℤ, while

    0 < D_N R_N ≤ Σ_{k≥1} 1/(4N+2)^k = 1/(4N+1) → 0.
Note from Claude Sonnet 5

Tweet by Costa (@Clambr0) saying a five-line, Lean-verified proof shows the Erdős-asked series sum of n!/(2n)! is irrational (extended to a family), and that his AI then claimed transcendence for a wider affine family, linking a GitHub repo for Erdős Problem 270; below, the start of the thread with the typeset irrationality proof. Posted 2026-08-22.

mathematicserdos problemsirrationalitytranscendenceleanai math resultsformal verification

shako @shakoistsLog

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shako @shakoistsLog · 12h
could you just.... like.... could you make an AI that just keeps making novel Erdos-style problems, then solving them indefinitely? Like... how deep is the well of mathematics?

[quoted tweet]
rohit @krishnanrohit · 12h
I hope AI can actually create more Erdos problems by 2028 x.com/sir_lemmings/s...

10 replies, 1 repost, 98 likes, 4.6K views

Shannon Sa... @max_papercl... · 10h
adversarial training, but it's just a pair of agents where 1 is creating Erdos problems & the other is solving them
Note from Claude Sonnet 5

Tweet from shako (@shakoistsLog) musing whether an AI could generate and solve novel Erdos-style math problems indefinitely, quote-tweeting rohit (@krishnanrohit) hoping AI creates more Erdos problems by 2028, with a reply from Shannon Sa... (@max_papercl...) suggesting an adversarial pair of agents, one generating problems and one solving them.

ai mathematicserdos problemsadversarial training

Dan Robinson @danrobinson

— reposted by Sichu Lu, quote-tweeting @Qiaoqiao2001 (Shouqiao Wang)

[Reposted by: Sichu Lu] @danrobinson (Dan Robinson) — 12:05 PM · Jul 22, 2026 · 67.1K Views The models are increasingly good at Erdos problems, but some experts like @Qiaoqiao2001 seem to have exceptional skill at prompting We sponsored him to see how far he could go He solved six this week, including one that had been studied by Terence Tao Here's how he did it: > QUOTED: @Qiaoqiao2001 (Shouqiao Wang) — 5h > I solved 6 open Erdős problems in 5 days, using @OpenAI GPT-5.6 Sol. > > I have a math background, but the Codex workflow I used does not require deep mathematical knowledge.... [truncated]
Note from Claude Sonnet 5

Tweet describing a sponsored effort where a prompting specialist (Shouqiao Wang) reportedly solved six open Erdős problems in a week using OpenAI's GPT-5.6 Sol via a Codex-based workflow, including one problem previously studied by Terence Tao.

twitterai for mathopenaierdos problemsprompting