The Von Staudt-Clausen Theorem describes the fractional part of a Bernoulli number, stating that adding the reciprocal of every prime p for which p minus one divides the Bernoulli number's even index to that Bernoulli number always produces an integer. Named for Karl von Staudt and Thomas Clausen, who proved it independently, it explains why the denominators of the Bernoulli numbers are exactly the product of such primes.
Facts
StatementFor a positive integer n, adding 1/p to the Bernoulli number B(2n) for every prime p such that p minus 1 divides 2n produces an integer, so the denominator of B(2n) is the product of all such primes. 1 Classification
Statement Form Connections
Has Statement Form
In Branch
Source Von Staudt-Clausen Theorem (Wikipedia)
Proved By
Source Von Staudt-Clausen Theorem (Wikipedia)
Source Von Staudt-Clausen Theorem (Wikipedia)
Sources
1. Von Staudt-Clausen Theorem (Wikipedia)
Wikimedia Foundationlead section, first sentence
the von Staudt-Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840)
lead section, first sentence, parenthetical years
found independently by Karl von Staudt (1840) and Thomas Clausen (1840)
- In Branch: Number Theory, Lead sentence
View the Source Von Staudt-Clausen Theorem (Wikipedia)
Proved By: Thomas Clausen, Lead sentence, attribution
found independently by Karl von Staudt (1840) and Thomas Clausen (1840)
Proved By: Karl von Staudt, Lead sentence, attribution
found independently by Karl von Staudt (1840) and Thomas Clausen (1840)
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.