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

**(28)**

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

**(29)**

whenever and .

For any positive integers *t* and *n*, we have

**(30)**

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 .