Problem Statement
If $1<k<n-1$ then $\binom{n}{k}$ is divisible by a prime $p<n/2$ (except $\binom{7}{3}=5\cdot 7$).
Categories:
Number Theory Binomial Coefficients
Progress
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].
Source: erdosproblems.com/384 | Last verified: January 14, 2026