A conjecture of Erdős and Selfridge. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever $n>k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$. They have proved the weaker inequality $p\ll n/k^c$ for some constant $c>0$.

Discussed in problem B31 and B33 of Guy's collection [Gu04] - there Guy credits Selfridge with the conjecture that if $n> 17.125k$ then $\binom{n}{k}$ has a prime factor $p\leq n/k$.

Stronger forms of this conjecture are [1094] and [1095].

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This page was last edited 28 October 2025.