Erdős Problem #397 — Equal Products of Central Binomial Coefficients

Erdős Problem $397$ asks whether products of distinct central binomial coefficients can coincide only finitely often. The answer is negative: for every integer $k\geq 3$, explicit disjoint index sets give a distinct identity.

Known results

• The family $A_k=\{k,2k-2,8k^2-8k+2\}$ and $B_k=\{k-1,2k,8k^2-8k+1\}$ satisfies the required identity for every $k\geq 3$.
• The verification reduces to $\binom{2x}{x}/\binom{2x-2}{x-1}=2(2x-1)/x$.
• The construction is essentially identical to a problem from the $2012$ Chinese Team Selection Test for the IMO.

Recent corroboration

Aletheia and, independently, GPT-5.2 Pro with Aristotle produced the same construction; the arXiv case study gives a complete verification. Human auditors later identified an earlier literature source, so the AI discovery is not novel, but the mathematical resolution is established.

Current status (as of February 2026): The problem is resolved negatively; infinitely many identities are established, while the AI-assisted rediscovery is not original.