# 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.

where *p* is a prime, *h* and *k* are positive even integers not divisible by *p*−1 and the numbers *B*_{h} are Bernoulli numbers.

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

where φ(*p*^{a+1}) is the Euler totient function, evaluated at *p*^{a+1} and *a* is a non negative integer. At *a* = 0, the expression takes the simpler form, as seen above.
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 continuous, so can be extended by continuity to all *p*-adic integers.