Erdős Problem #384

If \(1<k<n-1\) then \(\binom{n}{k}\) is divisible by a prime \(p\leq n/2\) (except \(\binom{7}{3}=\binom{7}{4}=5\cdot 7\)). A conjecture of Erdős and Selfridge [ErGr80]. 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 \(p<n/k\). Discussed in problem B31 and B33 of Guy's collection [Gu04]. Stronger forms of this conjecture are [1094](https://www.erdosproblems.com/1094) and [1095](https://www.erdosproblems.com/1095). Ecklund's theorem is stated for \(n \geq 2k\) with the single exception \(\binom{7}{3}\); by the symmetry \(\binom{n}{k} = \binom{n}{n-k}\) this is the form above, where \(\binom{7}{4}\) is the second exception. The prime bound is \(p \leq n/2\) (that is, \(2p \leq n\)); see `erdos_384.variants.strict` for the strict inequality.