Von Staudt–Clausen theorem

For each positive integer n, the denominator of the Bernoulli number B₂ₙ in lowest terms is the product of primes p such that p−1 divides 2n.

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Bernoulli numbers: A sequence of rational numbers that appears in power sums, series expansions, and formulas in number theory. The theorem specifies the denominators of the even-indexed members of this sequence.
Rational number: A number expressible as the ratio of two integers with a nonzero denominator. Bernoulli numbers are rational, so their reduced denominators are defined.
Karl von Staudt: A German mathematician whose work included a 1840 result on Bernoulli-number denominators. His independent result supplied one half of the theorem now bearing his name.
Faulhaber's formula: A formula expressing sums of powers of consecutive integers using Bernoulli numbers. The theorem identifies prime denominators in the coefficients used for power sums.
Euler's theorem: A result stating that powers of an integer coprime to a modulus are congruent to one modulo that modulus at a specified exponent. It also uses prime-related divisibility, but does not determine Bernoulli denominators.
Congruence: A relation stating that two integers have the same remainder upon division by a specified modulus. The theorem is commonly expressed as a congruence modulo integers.
Integer: A whole number that is zero, positive, or negative. The theorem’s index and divisibility condition are stated using integers.
Thomas Clausen: A Danish mathematician who independently established the Bernoulli-denominator result in 1840. His proof independently established the same denominator formula.
Riemann zeta function: A complex function initially defined by a series over reciprocal powers of positive integers. Its values at negative even integers are expressed using Bernoulli numbers.
Numerator: The part of a fraction written above the fraction bar. The theorem determines denominators, not the often more intricate prime factors of Bernoulli numerators.