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Claim. The answer to Problem 397 is no. Tony Feng and twenty-three coauthors, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401, first posted 29 January 2026 and carded as [[../library/distance_problems/feng_2026_semi_autonomous_mathematics_discovery_gemini_case/_index|Feng et al. 2026]] (version 3 of 5 February 2026, linked above), state in Section 4.1 as Theorem 4 that there are infinitely many pairs of disjoint finite sets $A,B$ of positive integers with

$$\prod_{m\in A}\binom{2m}{m}=\prod_{n\in B}\binom{2n}{n}.$$

The proof is an explicit family: for every integer $k\ge3$, $A_k=\{k,\,2k-2,\,8k^2-8k+2\}$ and $B_k=\{k-1,\,2k,\,8k^2-8k+1\}$, two disjoint three-element sets, and the ratio of the two products telescopes to $1$ through the identity $C_x/C_{x-1}=2(2x-1)/x$ for consecutive central binomial coefficients $C_x=\binom{2x}{x}$; different $k$ give different pairs. Writing $a=k-1$ turns the family into the one Neel Somani posted in the site's discussion thread on 11 January 2026, recorded with its verification on Somani's page.

Claimant. The paper attributes the solution to Aletheia, a research agent built on Gemini Deep Think, in a deployment the authors date to 2 through 9 December 2025; its Remark 4.1 says that the printed solution was edited from the raw model output mainly for length, that a long computational verification of the identity was shortened, and that all mathematical content of the printed solution was present in the raw output. The paper lists the result among its independent rediscoveries. It writes that the same family was afterwards found independently by GPT-5.2 Pro together with Aristotle, that the problem was later found to be essentially Problem 3 of Day 1 of the 2012 China Team Selection Test (Art of Problem Solving topic 469502, post 2628490), and that the authors cede priority. The occurrence in the 2012 test is recorded here as the paper reports it.

Standing. The claim is claimed. The site's problem page (last edited 12 January 2026) credits the disproof to Somani and does not mention this report; no reviewer has accepted the paper's account of the result, and the preprint has no journal publication. The problem's standing derives from Somani's accepted page, and this page records the same answer by an independent route.

Depends on. No page of this wiki.