Computing the Greatest Common Divisor of Binomial Coefficients \binommnmk

2026/06/30 by Chai Wah Wu
Mathematics · #math.NT #msc:11B65 #msc:11A05 #msc:11A41 #acm:11B65 #acm:11A05 #acm:11A41

paper · pdf

6 pages

arxiv created 2026/08/04 · arxiv updated 2026/08/06

Abstract

The greatest common divisor (GCD) of \binom2n2k for 1≤ k<n is known to be some power of 2 times the product of all odd primes p such that 2n=pⁱ+p^j. We complete the analysis of this GCD by showing that this power of 2 is either 1 or 0 and relates it to Mersenne primes. We also show how to efficiently compute GCD \\binommnmk: 1≤ k<n\ when n and m satisfy certain conditions.

Citations

• On the Greatest Common Divisor of Binomial Coefficients n \choose q, n \choose 2q, n \choose 3q, …
• The Cayley plane and String bordism
• Primary cyclotomic units and a proof of Catalans conjecture