The Bernoulli numbers, , and the Bernoulli polynomials, , are defined by the power series
so that and . Some useful facts, that follow straight from these definitions, are that each is a rational number, if n is odd and , and
for all integers .
In 1840 Clausen and Von Staudt showed that the denominator of (n even) is precisely the product of those primes p for which p-1 divides n; and further that for each such p (actually one also has ). In 1851 Kummer showed that for any even integers m and n, satisfying and ; and one can use this in showing that
whenever and .
For any positive integers t and n, we have
and Kummer's congruences for Bernoulli polynomials. Consequently, if m divides up+v, for given integers , then, by combining the identity
with (30), we obtain
for primes provided .