Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$

Chai Wah Wu

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arXiv · 2606.20940

Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$

Chai Wah Wu

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Abstract

The greatest common divisor (GCD) of $\binom{2n}{2k}$ for $1\leq k<n$ is known to be some power of 2 times the product of all odd primes p such that $2n=p^i+p^j$. 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 $GCD \{\binom{mn}{mk}: 1\leq k<n\}$ when n and m satisfy certain conditions.

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Chai Wah Wu. 2026-06-18. Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$. https://arxiv.org/abs/2606.20940

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