Erdős Problem #397

Aristotle and GPT-5.2 Pro produced a solution (10 Jan 2026), but a comparable result was already in the literature: China TST competition (2012). 3 AI contributions are recorded in all.

Problem
Erdős problem #397 (number theory, binomial coefficients); current status: disproved (Lean)
Credit
Aristotle and GPT-5.2 Pro · AI on its own; related literature found afterwards
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2026-01-10 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PRIOR
ARTPer Tao's AI-contributions wiki

The problem

Are there only finitely many solutions to

∏i(2mimi)=∏j(2njnj)\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}

with the mi,njm_i,n_j distinct?

Statement quoted from erdosproblems.com/397, maintained by Thomas Bloom. Original source: [ErGr80,p.77]. Current status there: disproved (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 Literature
10 Jan, 2026 AI on its own; related literature found afterwards Aristotle, GPT-5.2 Pro 🟢 Full solution (Lean) 🟢 China TST competition (2012)
10 Jan, 2026 Literature search ChatGPT Deep research 🟢 Full solution found —
2 Feb, 2026 AI on its own; related literature found afterwards Aletheia 🟢 Full solution 🟢 China TST competition (2012)

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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