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Claim. There is an $n_0$ such that $\binom{2n}{n}$ is not squarefree for every $n\ge n_0$. This is the theorem of A. Sárközy, On divisors of binomial coefficients, I, Journal of Number Theory 20 (1985), no. 1, 70–80; the journal record dates the issue to February 1985 without a day, so this page is dated to the first of that month. The paper is not held in the library. As Velammal describes it, the argument uses Jutila's estimates for exponential sums over primes to show that some prime $p$ has $p^2\mid\binom{2n}{n}$ once $n$ is large, and Granville and Ramaré say that their own proof follows it in turning the question into exponential sums; the estimates give no value of $n_0$.

Covers. The statement of Problem 175 for all sufficiently large $n$, with a threshold the paper does not make explicit. The full range $n\ge5$ was proved a decade later, independently, by Velammal and by Granville and Ramaré, each making the exponential-sum bounds explicit and checking the remaining range; their results are on Velammal's and Granville and Ramaré's claim pages.

Depends on. No page of this wiki.

Acceptance. The paper appeared in the Journal of Number Theory, a refereed journal, and Thomas Bloom, the site's curator, credits it on the problem page (last edited 8 February 2026) as the first proof for all large $n$. Both later full proofs cite it as the result whose bounds they make explicit.