Von Staudt–Clausen theorem
Encyclopedia
In number theory
, the von Staudt–Clausen theorem is a result determining the fractional part
of Bernoulli number
s, found independently by
and .
Specifically, if we add 1/p to Bn for every prime
p such that p − 1 divides n, we obtain an integer.
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers Bn as the product of all primes p such that p − 1 divides n; consequently the denominators are square-free
and divisible by 6.
These denominators are
The second part of the von Staudt–Clausen theorem is a representation of the Bernoulli numbers. This representation is given for the first few nonzero Bernoulli numbers in the next table.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the von Staudt–Clausen theorem is a result determining the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
of Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s, found independently by
and .
Specifically, if we add 1/p to Bn for every prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p such that p − 1 divides n, we obtain an integer.
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers Bn as the product of all primes p such that p − 1 divides n; consequently the denominators are square-free
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
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, ...
Formulation
The von Staudt–Clausen theorem has two parts. The first one describes how the denominators of the Bernoulli numbers can be computed. Paraphrasing the words of Clausen it can be stated as:“The denominator of the 2nth Bernoulli number can be found as follows: Add to all divisors of 2n, 1, 2, a, a
The second part of the von Staudt–Clausen theorem is a representation of the Bernoulli numbers. This representation is given for the first few nonzero Bernoulli numbers in the next table.
| Von Staudt–Clausen representation of Bn | ||
| B0 | = | 1 |
| B1 | = | − 1/2 |
| B2 | = | 1 − 1/2 − 1/3 |
| B4 | = | 1 − 1/2 − 1/3 − 1/5 |
| B6 | = | 1 − 1/2 − 1/3 − 1/7 |
| B8 | = | 1 − 1/2 − 1/3 − 1/5 |
| B10 | = | 1 − 1/2 − 1/3 − 1/11 |