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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 one can use this in showing that
whenever and .
Recently Almkvist and Meurman  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 .