# Kummer’s congruence

Let $B_{k}$ denote the $k$th Bernoulli number:

 $B_{0}=1,\quad B_{1}=-\frac{1}{2},\quad B_{2}=\frac{1}{6},\quad B_{3}=0,\quad B% _{4}=-\frac{1}{30},\ldots,\ B_{10}=\frac{5}{66},\ldots$

In fact, $B_{k}=0$ for all odd $k\geq 3$, so we will only consider $B_{k}$ for even $k$. The following congruence is due to Ernst Eduard Kummer:

###### Theorem (Kummer’s congruence).

Let $p$ be a prime. Suppose that $k\geq 2$ is an even integer which is not divisible by $(p-1)$. Then the quotient $B_{k}/k$ 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 $(p-1)\nmid k$ and $k\equiv h\mod(p-1)$ then

 $\frac{B_{k}}{k}\equiv\frac{B_{h}}{h}\mod p.$

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:

 $\frac{B_{4}}{4}=\frac{-\frac{1}{30}}{4}=-\frac{1}{120}\equiv 6\mod 7$

If we pick $h=10$ (so that $10\equiv 4\mod(p-1)$) then:

 $\frac{B_{10}}{10}=\frac{\frac{5}{66}}{10}=\frac{1}{132}\equiv 6\mod 7$

which is what the theorem predicted.

Title Kummer’s congruence KummersCongruence 2013-03-22 15:12:01 2013-03-22 15:12:01 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 11B68 Kummer congruence CongruenceOfClausenAndVonStaudt IntegralElement OddBernoulliNumbersAreZero