Problem 175

Show that, for any $n\geq 5$ , the binomial coefficient $\binom{2n}{n}$ is not squarefree.

Source: erdosproblems.com/175

Number theory
Binomial coefficients

Wiki page

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SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN) (page last edited 8 February 2026) and credits Sárközy for all sufficiently large $n$ and, independently, Granville and Ramaré and Velammal for every $n\ge5$ ; the three results are recorded on the claim pages Sárközy 1985 (partial), Velammal 1995 and Granville and Ramaré 1996. The Lean qualifier refers to Boris Alexeev's formalization of the Granville–Ramaré argument in his repository of formalized Erdős problems, which this corpus has not built; the two full proofs are refereed. The standing in the frontmatter derives from the claim pages.

ErKo99: Erdős, Paul and Kolesnik, Grigori, Prime power divisors of binomial coefficients. Discrete Math. (1999), 101-117.
GrRa96: Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43 (1996), no. 1, 73-107. Library home: Granville and Ramaré 1996.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B33 "Largest divisor of a binomial coefficient", printed p. 135, where the book states the conjecture and reports the Sárközy, Sander and Granville and Ramaré results. Library home: Guy 2004.
Sa85: Sárközy, A., On divisors of binomial coefficients, I. Journal of Number Theory 20 (1985), no. 1, 70-80.
Sa92: Sander, J. W., Prime power divisors of binomial coefficients. J. Reine Angew. Math. 430 (1992), 1-20. Library home: Sander 1992.
Sa92b: Sander, J. W., On prime divisors of binomial coefficients. Bull. London Math. Soc. (1992), 140-142.
Sa95: Sander, J. W., On the order of prime powers dividing $\binom {2n}n$ . Acta Math. (1995), 85-118.
Ve95: Velammal, G., Is the binomial coefficient $\binom {2n}n$ square free?. Hardy-Ramanujan J. 18 (1995), 23-45. Library home: Velammal 1995.

Problem 175

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