Classically, the Bernoulli numbers are defined by . These numbers are rational and, for odd , . For even , and we can write uniquely , where are relatively prime integers and . The following theorem concerning the numerators is due to von Staudt (1845): “Let be even, and a prime with . If divides (), then divides , too.”
A great number of mathematicians have given various proofs of this theorem since. The author presents quite a new proof based on the notion of “cyclotomic” Bernoulli numbers defined as follows
where is a primitive th root of unity for .