Kummer’s congruence

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 congruenceMathworldPlanetmathPlanetmathPlanetmath is due to Ernst Eduard Kummer:

Theorem (Kummer’s congruence).

Let p be a prime. Suppose that k2 is an even integer which is not divisible by (p-1). Then the quotient Bk/k is p-integral, that is, as a fraction in lower terms, p does not divide its denominator. Furthermore, if h is another even integer with (p-1)k and khmod(p-1) then


The interested reader should see also the congruence of Clausen and von Staudt for a similar result. As an example of Kummer’s congruence, let p=7 and k=4. Then:


If we pick h=10 (so that 104mod(p-1)) then:


which is what the theorem predicted.

Title Kummer’s congruence
Canonical name KummersCongruence
Date of creation 2013-03-22 15:12:01
Last modified on 2013-03-22 15:12:01
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 11B68
Synonym Kummer congruence
Related topic CongruenceOfClausenAndVonStaudt
Related topic IntegralElement
Related topic OddBernoulliNumbersAreZero