In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer. Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function.
Statement The simplest form of Kummer's congruence states that
B h h ≡ B k k ( mod p ) whenever h ≡ k ( mod p − 1 ) {\displaystyle {\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 ) B h h ≡ ( 1 − p k − 1 ) B k k ( mod p a + 1 ) {\displaystyle (1-p^{h-1}){\frac {B_{h}}{h}}\equiv (1-p^{k-1}){\frac {B_{k}}{k}}{\pmod {p^{a+1}}}}
whenever
h ≡ k ( mod φ ( p a + 1 ) ) {\displaystyle 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 Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers Bernoulli number § The Kummer theorems
References
Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3, MR 0754003