On the Greatest Common Divisor of Binomial Coefficients ${n \choose q}, {n \choose 2q}, {n \choose 3q}, \dots$

TL;DR

This paper explores the greatest common divisor of specific binomial coefficients, generalizing known results and providing a concise proof of these properties related to prime factors and binomial coefficient indices.

Contribution

It offers a new, concise proof of a generalization concerning the gcd of binomial coefficients with indices multiple of a fixed number.

Findings

The gcd of binomial coefficients inom{n}{q}, inom{n}{2q}, \u2026 is characterized by prime factors related to the sum of prime powers.

A generalization of known gcd results for binomial coefficients is established.

The paper clarifies the prime factor structure of these gcds in relation to binomial coefficient indices.

Abstract

Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\binom n1,\binom n2,\dots,\binom n{n-1}$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that the greatest common divisor of the binomial coefficients $\binom{2n}2,\binom{2n}4,\dots,\binom{2n}{2n-2}$ equals (a certain power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $0\le i\le j$. This note gives a concise proof of a tidy generalization of these facts.