Kummer's congruence explained

In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by .

used Kummer's congruences to define the p-adic zeta function.

Statement

The simplest form of Kummer's congruence states that

(B _{h})/(h) \equiv (B _{k})/(k) \pmod p 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 B_h are Bernoulli numbers.

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

( 1- p ^{h-1} ) (B _{h})/(h) \equiv ( 1- p ^{k-1} ) (B _{k})/(k) \pmod {p ^{a+1} }wheneverh\equiv k\pmod {\varphi(p ^{a+1} )}

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.

See also

• Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers