Kummer's congruence

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Short description: Result in number theory showing congruences involving Bernoulli numbers

In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851).

(Kubota Leopoldt) used Kummer's congruences to define the p-adic zeta function.

Statement

The simplest form of Kummer's congruence states that

[math]\displaystyle{ \frac{B_h}{h}\equiv \frac{B_k}{k} \pmod p \text{ whenever } h\equiv k \pmod {p-1} }[/math]

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

[math]\displaystyle{ (1-p^{h-1})\frac{B_h}{h}\equiv (1-p^{k-1})\frac{B_k}{k} \pmod {p^{a+1}} }[/math]

whenever

[math]\displaystyle{ h\equiv k\pmod {\varphi(p^{a+1})} }[/math]

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

References