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Contents Next: Generalization of Morley's Up: Arithmetic Properties of Binomial Previous: Sums of Binomial

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

.

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Contents Next: Generalization of Morley's Up: Arithmetic Properties of Binomial Previous: Sums of Binomial