Problem 379

Let $S(n)$ denote the largest integer such that, for all $1\leq k<n$ , the binomial coefficient $\binom{n}{k}$ is divisible by $p^{S(n)}$ for some prime $p$ (depending on $k$ ). Is it true that

\limsup S(n)=\infty?

Source: erdosproblems.com/379

Number theory
Binomial coefficients

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN) (page last edited 12 January 2026) and credits a proof worked out in its discussion thread by Cambie, Kovač and Tao in August 2025, recorded on its claim page (Cambie, Kovač and Tao, 2025); the Lean qualifier refers to Tao's formalization of that proof in his analysis repository, which this corpus has not built. There is no refereed write-up. A second, independent proof, produced by the Seed-Prover 1.5 system and posted in the thread by Zheng Yuan on 21 December 2025, is an accepted claim on formal evidence on its own page: this corpus built a version of its Lean code and audited its statement. The standing in the frontmatter derives from the claim pages.

Problem 379

Status

Proof claims

Discussion