Problem 387

Is there an absolute constant $c>0$ such that, for all $1\leq k< n$ , the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$ ?

Source: erdosproblems.com/387

Number theory
Binomial coefficients

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SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem SOLVED, crediting Bui, Naprienko, Pratt and Zaharescu with the negative answer. The standing derived from the claim pages is solved, disproved: no constant $c>0$ works, by the accepted claim Bui, Naprienko, Pratt and Zaharescu 2026.

BNPZ26: H. Bui, S. Naprienko, K. Pratt, and A. Zaharescu, Binomial coefficients with divisors avoiding an interval. arXiv:2605.21221 (2026).
Er78g: Erdős, Pál, On prime factors of binomial coefficients. II. Mat. Lapok (1978/82), 307-316.
Fa66: Faulkner, M., On a theorem of Sylvester and Schur. J. London Math. Soc. (1966), 107-110.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B33 "Largest divisor of a binomial coefficient", printed p. 134: Erdős's conjecture that $\binom nk$ has a divisor between $cn$ and $n$ for any $c<1$ and $n$ sufficiently large. Library home: Guy 2004.
Sc58: Schinzel, A., Sur un problème de P. Erdős. Colloq. Math. (1958), 198-204.

Problem 387

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