 Von Staudt–Clausen theorem

For $n\ge 1$, the von Staudt-Clausen theorem statesConsequently the positive denominator of the Bernoulli number $B_{2n}$ in lowest terms is exactly the squarefree product of the primes satisfying $p-1\mid 2n$. This gives its denominator without computing its numerator.^[ref]

For $B_4$, the relevant primes are $2,3,5$. Their reciprocals sum to $31/30$, and $B_4=-1/30$, so the displayed expression is $1$. The theorem concerns even positive indices; $B_1=-1/2$ is handled separately. It explains the recurring small prime factors in denominators of formulas for sums of powers.

The Von Staudt–Clausen theorem is a result in the field of number theory, particularly concerning the theory of continued fractions and the approximation of numbers. The theorem provides a way to express a specific class of numbers, notably the values of certain mathematical constants, as a sum involving continued fractions.