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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 .
Recently Almkvist and Meurman [1] observed that
is an integer for any rational number .
For any positive integers **t** and **n**, we have

From this, it is easy to deduce analogues for Bernoulli polynomials
of the Von Staudt--Clausen Theorem and of Kummer's congruences.
We shall assume that
**m** divides **up+v**, for given integers .
By (8.3) we have

for , provided with .
Thus, under these conditions,
and so, from (8.3), we obtain
for primes provided .