Computing the Greatest Common Divisor of Binomial Coefficients $\left(\genfrac{}{}{0px}{}{mn}{mk}\right)$

The greatest common divisor (GCD) of $\left(\genfrac{}{}{0px}{}{2n}{2k}\right)$ for $1\le k<n$ 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$\left(\genfrac{}{}{0px}{}{mn}{mk}\right):1\le k<n$ when n satisfies certain conditions.