As is so well known, for any fixed n, the sum, over all m, of the binomial coefficients , is exactly 2 to the power n. What is less well-known is that the sum of the binomial coefficients, over all m in certain fixed residue classes, sometimes satisfy certain surprising congruences:
In 1876 Hermite showed that if n is odd then the sum of the binomial coefficients , over those positive integers m that are divisible by p-1, is divisible by p.
In 1899 Glaisher generalized this by showing that for any given prime p and integers , we have
for all positive integers .
In 1953 Carlitz generalized Hermite's Theorem to prime powers: If divides n, with and , then
In 1913 Fleck gave the related result that for any given prime p and integers , we have
In 1965 Bhaskaran showed that if p is an odd prime then p+1 divides n if and only if