Kummer's congruence

In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851).

Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function.

Statement

The simplest form of Kummer's congruence states that

\frac{B_h}{h}\equiv \frac{B_k}{k} \bmod p \text{ whenever } h\equiv k \bmod p-1

where p is a prime, h and k are positive even integers not divisible by p−1, and the numbers Bh are Bernoulli numbers.

More generally if h and k are positive even integers not divisible by p − 1, then

(1-p^{h-1})\frac{B_h}{h}\equiv (1-p^{k-1})\frac{B_k}{k} \bmod p^{a%2B1}

whenever

h\equiv k\bmod (p-1)p^a

The two sides of the Kummer congruence are essentially values of the p-adic zeta function, and the Kummer congruences imply that the p-adic zeta function for negative integers is continuously, so can be extended by continuity to all p-adic integers.

See also

References