Editing Bernoulli number (section)

=== Von Staudt–Clausen theorem ===
{{main|Von Staudt–Clausen theorem}}
The von Staudt–Clausen theorem was given by [[Karl Georg Christian von Staudt]]{{r|vonStaudt1840}} and [[Thomas Clausen (mathematician)|Thomas Clausen]]{{r|Clausen1840}} independently in 1840. The theorem states that for every {{math|''n'' > 0}},
: <math> B_{2n} + \sum_{(p-1)\,\mid\,2n} \frac1p</math>
is an integer. The sum extends over all [[prime number|primes]] {{math|''p''}} for which {{math|''p'' − 1}} divides {{math|2''n''}}.

A consequence of this is that the denominator of {{math|''B''<sub>2''n''</sub>}} is given by the product of all primes {{math|''p''}} for which {{math|''p'' − 1}} divides {{math|2''n''}}. In particular, these denominators are [[square-free]] and divisible by&nbsp;6.