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.