Erdős Problem #684

The problem

For $0\leq k\leq n$ write
$\binom{n}{k} = uv$
where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$ .

Let $f(n)$ be the smallest $k$ such that $u>n^2$ . Give bounds for $f(n)$ .

Statement quoted from erdosproblems.com/684, maintained by Thomas Bloom. Original source: [Er79d]. Current status there: open.

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	Humans	Outcome
19 Jan, 2026	AI working with humans	GPT-5.2 Thinking	Quanyu Tang	🟡 Partial result
31 Mar, 2026	AI on its own	OpenAI internal model	—	🟡 Partial result
2 Apr, 2026	AI working with humans	GPT-5.4 Thinking	Nat Sothanaphan	🟡 Partial result

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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