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

where .

In 1965 Bhaskaran showed that if *p* is an odd prime then *p+1* divides *n*
if and only if

for .