The simplest form of Kummer's congruence states that : \frac{B_h}{h}\equiv \frac{B_k}{k} \pmod p \text{ whenever } h\equiv k \pmod {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} \pmod {p^{a+1}} whenever : h\equiv k\pmod {\varphi(p^{a+1})} where φ(
pa+1) is the
Euler totient function, evaluated at
pa+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. ==See also==