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
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Von Staudt-Clausen Theorem (Wikipedia)
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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
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