Kummer's congruence

The simplest form of Kummer's congruence states that

:$\backslash frac\{B\_h\}\{h\}\backslash equiv\; \backslash frac\{B\_k\}\{k\}\; \backslash pmod\; p\; \backslash text\{\; whenever\; \}\; h\backslash equiv\; k\; \backslash pmod\; \{p-1\}$

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

:$(1-p^\{h-1\})\backslash frac\{B\_h\}\{h\}\backslash equiv\; (1-p^\{k-1\})\backslash frac\{B\_k\}\{k\}\; \backslash pmod\; \{p^\{a+1\}\}$

whenever

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

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
• {{slink|Bernoulli number#The Kummer theorems}}

• {{Citation | last1=Koblitz | first1=Neal | author1-link=Neal Koblitz | title=p-adic Numbers, p-adic Analysis, and Zeta-Functions | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics, vol. 58 | isbn=978-0-387-96017-3 | mr=754003 | year=1984}}
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Category:Theorems in number theory

Category:Modular arithmetic