In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the von Staudt–Clausen theorem is a result determining the
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than , called ''floor'' of or \lfloor x\rfloor. Then, the fractional ...
of
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 function ...
s, found independently by
and .
Specifically, if is a positive integer and we add to the Bernoulli number for every
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
such that divides , then we obtain an integer; that is,
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers as the product of all primes such that divides ; consequently, the denominators are
square-free {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
...
and divisible by 6.
These denominators are
: 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... .
The sequence of integers
is
: 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... .
Proof
A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is:
:
and as a corollary:
:
where are the
Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
.
Furthermore the following lemmas are needed:
Let be a prime number; then
1. If divides , then
:
2. If does not divide , then
:
Proof of (1) and (2): One has from
Fermat's little theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
a^p \equiv a \pmod p.
For example, if and , t ...
,
:
for .
If divides , then one has
:
for . Thereafter, one has
:
from which (1) follows immediately.
If does not divide , then after Fermat's theorem one has
:
If one lets , then after iteration one has
:
for and .
Thereafter, one has
:
Lemma (2) now follows from the above and the fact that for .
(3). It is easy to deduce that for and , divides .
(4). Stirling numbers of the second kind are integers.
Now we are ready to prove the theorem.
If is composite and , then from (3), divides .
For ,
:
If is prime, then we use (1) and (2), and if is composite, then we use (3) and (4) to deduce
:
where is an integer, as desired.
[T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.]
See also
*
Kummer's congruence
References
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External links
*
{{DEFAULTSORT:Von Staudt Clausen Theorem
Theorems in number theory