On the Greatest Common Divisor of Binomial Coefficients $\left(\genfrac{}{}{0ex}{}{n}{q}\right),\left(\genfrac{}{}{0ex}{}{n}{2q}\right),\left(\genfrac{}{}{0ex}{}{n}{3q}\right),\dots$

Abstract

Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{1}\right),\left(\genfrac{}{}{0ex}{}{n}{2}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{n-1}\right)$ 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 $\left(\genfrac{}{}{0ex}{}{2n}{2}\right),\left(\genfrac{}{}{0ex}{}{2n}{4}\right),\dots ,\left(\genfrac{}{}{0ex}{}{2n}{2n-2}\right)$ 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.

arXiv comment3 pages; thanks to Günter Ziegler for pointing out Balak Ram's 1909 proof of Theorem 1; typos corrected in first remark on p.2

References in corpus (1)

• The Cayley plane and String bordism