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 Karl von Staudt (1840) and Thomas Clausen (1840).

Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B_2n for every prime p such that p − 1 divides 2n, we obtain an integer, i.e., ${\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} .}$

This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B_2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6.

These denominators are

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

and as a corollary:

where ${\displaystyle S(n,j)\!}$ are the Stirling numbers of the second kind.

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