Computing the Greatest Common Divisor of Binomial Coefficients binomial(mn,mk)

The greatest common divisor (GCD) of binomial(2n, 2k) for 1≤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 its $2$-adic valuation is at most $1$ and relates it to Mersenne primes. We also show how to efficiently compute GCD {C(mn,mk) : 1 ≤ k < n} when n and m satisfies certain conditions.