On the Greatest Common Divisor of $\binom{qn}{q}, \binom{qn}{2q},\dots, \binom{qn}{qn-q}$

Carl McTague

Published 2015-10-22Version 1

Every binomial coefficient aficionado knows that $\mathrm{GCD}_{0<k<n}\binom n k$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that $\mathrm{GCD}_{0<k<n} \binom{2n}{2k}$ equals (a power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $i\le j$. This note gives a concise proof of a tidy generalization of these facts.