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Claim. The statement of Problem 175 holds: for every $n>4$ the central binomial coefficient $\binom{2n}{n}$ is not squarefree. This is the main result of G. Velammal, Is the binomial coefficient $\binom{2n}{n}$ square free?, Hardy-Ramanujan Journal 18 (1995), 23–45, DOI 10.46298/hrj.1995.132, carded at velammal_1995_is_binomial_coefficient_squarefree; the journal's record dates the publication 1 January 1995. Writing $\binom{2n}{n}=s(n)^2q(n)$ with $q(n)$ squarefree, the paper's Theorem (p. 24) states that $\binom{2n}{n}$ is never squarefree for $n\ge2^{8000}$; its proof exhibits a prime $p$ with both $\{n/p\}$ and $\{n/p^2\}$ at least $1/2$, which forces $p^2\mid\binom{2n}{n}$, and the estimates come from Vaughan's identity and the theory of exponent pairs, in place of the bounds Sárközy had used for large $n$. The range $4<n<2^{8000}$ is settled by direct elementary checks. The proof is independent of Granville and Ramaré's, recorded on their claim page, which also carries the one Lean formalization of the problem, a development that names Granville and Ramaré as its informal authors and lists this paper among its mathematical sources.

Depends on. No page of this wiki.

Acceptance. The paper appeared in the Hardy-Ramanujan Journal, a refereed journal. A comment of 6 February 2026 in the site's discussion pointed out that the paper proves the full range, and Thomas Bloom, the site's curator, credits Velammal beside Granville and Ramaré on the problem page (last edited 8 February 2026), where the problem is marked proved.