Erdős Problem #457

Full solution by Aristotle and GPT-5.2 Pro, formalized in Lean (2 Mar 2026).

Problem
Erdős problem #457 (number theory); current status: proved (Lean)
Credit
Aristotle and GPT-5.2 Pro · AI on its own
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2026-03-02 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Is there some ϵ>0\epsilon>0 such that there are infinitely many nn where all primes p≤(2+ϵ)log⁡np\leq (2+\epsilon)\log n divide

∏1≤i≤log⁡n(n+i)?\prod_{1\leq i\leq \log n}(n+i)?

Statement quoted from erdosproblems.com/457, maintained by Thomas Bloom. Original source: [Er79d,p.78][ErGr80,p.91]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Outcome
2 Mar, 2026 AI on its own Aristotle, GPT-5.2 Pro 🟢 Full solution (Lean)

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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