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 B2n for every prime p such that p − 1 divides 2n, then we obtain an integer; that is,
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently, the denominators are square-free and divisible by 6.
is A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is: and as a corollary: where S(n,j) are the Stirling numbers of the second kind.
Furthermore the following lemmas are needed: Let p be a prime number; then 1.
If p – 1 does not divide 2n, then after Fermat's theorem one has If one lets ℘ = ⌊ 2n / (p – 1) ⌋, then after iteration one has for m = 1, 2, ..., p – 1 and 0 < 2n – ℘(p – 1) < p – 1.
For j = 3, If j + 1 is prime, then we use (1) and (2), and if j + 1 is composite, then we use (3) and (4) to deduce where In is an integer, as desired.