Von Staudt–Clausen theorem

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In number theory, the von Staudt–Clausen theorem is a result on the fractional part of Bernoulli numbers. Specifically, if we add 1/p to B_n for every prime p such that p − 1 divides n, we obtain an integer.

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

The result is named for Karl von Staudt (1798–1867) and Thomas Clausen (1801–1885), who independently discovered the result in 1840.