Kummer's congruence

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The simplest form of Kummer's congruence states that

${\displaystyle {\frac {B_{h}}{h}}\equiv {\frac {B_{k}}{k}}{\pmod {p}}{\text{ whenever }}h\equiv k{\pmod {p-1}}}$

where p is a prime, h and k are positive even integers not divisible by p−1 and the numbers B_h are Bernoulli numbers.

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

${\displaystyle (1-p^{h-1}){\frac {B_{h}}{h}}\equiv (1-p^{k-1}){\frac {B_{k}}{k}}{\pmod {p^{a+1}}}}$

whenever

${\displaystyle h\equiv k{\pmod {\varphi (p^{a+1})}}}$

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.

• Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers
• Bernoulli number § The Kummer theorems