congruence of Clausen and von Staudt

Let Bk denote the kth Bernoulli numberMathworldPlanetmathPlanetmath:


In fact, Bk=0 for all odd k3, so we will only consider Bk for even k. The following is a well-known congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath, due to Thomas Clausen and Karl von Staudt.

Theorem (Congruence of Clausen and von Staudt).

For an even integer k2,

Bk-p prime,(p-1)|k1pmod

where the sum is over all primes p such that (p-1) divides k. In other words, there exists an integer nk such that

Bk=nk-p prime,(p-1)|k1p.

For example:


Sometimes the theorem is stated in this alternative form:


For an even integer k2 and any prime p the productPlanetmathPlanetmath pBk is p-integral, that is, pBk is a rational numberPlanetmathPlanetmathPlanetmath t/s (in lowest terms) such that p does not divide s. Moreover:

pBk{-1modp, if (p-1) divides k;0modp, if (p-1) does not divide k.
Title congruence of Clausen and von Staudt
Canonical name CongruenceOfClausenAndVonStaudt
Date of creation 2013-03-22 15:11:58
Last modified on 2013-03-22 15:11:58
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 11B68
Synonym Staudt-Clausen theorem
Synonym von Staudt-Clausen theorem
Related topic KummersCongruence
Related topic OddBernoulliNumbersAreZero