Problem 1094

For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$ , with only finitely many exceptions.

Status

Open

No claim settles this problem.

ELS88: Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507-523.
ELS93: Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215-224.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B31 "Binomial coefficients", printed p. 130: Selfridge's conjecture that $\binom nk$ with $n\ge2k$ has a prime factor $p\le n/k$ whenever $n>17.125k$ , the slightly stronger conjecture that the least prime factor is at most $\max(n/k,17)$ apart from exactly four coefficients, whose least prime factors are $19$ , $19$ , $23$ and $29$ , and the Erdős--Selfridge function $g(k)$ . Library home: Guy 2004.
Se77: J. L. Selfridge, Some problems on the prime factors of consecutive integers. Notices Amer. Math. Soc. (1977), A456-457.