Kummer's congruence

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In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer.[1]

Kubota & Leopoldt (1964) used Kummer's congruences to define the p-adic zeta function.[2]

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}}} {\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}}}} {\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})}}} {\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

References