Von Staudt–Clausen theorem

In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by and .

Specifically, if is a positive integer and we add to the Bernoulli number for every prime such that divides , then we obtain an integer; that is,

This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers as the product of all primes such that divides ; consequently, the denominators are square-free and divisible by 6.

These denominators are 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... .

The sequence of integers is 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... .

Proof A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is:

and as a corollary:

where are the Stirling numbers of the second kind.

Furthermore the following lemmas are needed:

Let be a prime number; then

1. If divides , then

2. If does not divide , then

Proof of (1) and (2): One has from Fermat's little theorem,

for .

If divides , then one has

for . Thereafter, one has

from which (1) follows immediately.

If does not divide , then after Fermat's theorem one has

If one lets , then after iteration one has

for and .

Thereafter, one has

Lemma (2) now follows from the above and the fact that for .

(3). It is easy to deduce that for and , divides .

(4). Stirling numbers of the second kind are integers.

Now we are ready to prove the theorem.

If is composite and , then from (3), divides .

For ,

If is prime, then we use (1) and (2), and if is composite, then we use (3) and (4) to deduce

where is an integer, as desired.H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

Von Staudt–Clausen theorem

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