In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Kummer's congruences are some
congruences involving
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, found by .
used Kummer's congruences to define the
p-adic zeta function
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime nu ...
.
Statement
The simplest form of Kummer's congruence states that
:
where ''p'' is a prime, ''h'' and ''k'' are positive even integers not divisible by ''p''−1 and the numbers ''B''
''h'' are
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s.
More generally if ''h'' and ''k'' are positive even integers not divisible by ''p'' − 1, then
:
whenever
:
where φ(''p''
''a''+1) is the
Euler totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
, 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
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime nu ...
, 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
*
Von Staudt–Clausen theorem
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by
and .
Specifically, if ''n'' is a positive integer and we add 1/''p'' to the Bernoulli number ''B''2''n'' ...
, another congruence involving Bernoulli numbers
References
*
*
*{{Citation , last1=Kummer , first1=Ernst Eduard , title=Über eine allgemeine Eigenschaft der rationalen Entwicklungscoëfficienten einer bestimmten Gattung analytischer Functionen , url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002147319 , id={{ERAM, 041.1136cj , year=1851 , journal=Journal für die Reine und Angewandte Mathematik , issn=0075-4102 , volume= 41 , pages=368–372 , doi=10.1515/crll.1851.41.368
Theorems in number theory
Modular arithmetic