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Bernoulli numbers and polynomials.


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 .