## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  405.10011
Autor:  Erdös, Paul; Wagstaff, Samuel S.jun.
Title:  The fractional parts of the Bernoulli numbers. (In English)
Source:  Ill. J. Math. 24, 104-112 (1980).
Review:  It is proved that the fractional parts of the Bernoulli numbers B2 k are dense in the interval (0,1). Furthermore, for every positive integer k, the set of all m for which B2 m has the same fractional part as B2 k as positive asymptotic density. The second statement is proved via this result on divisibility by p-1: For each \epsilon > 0, there is a T = T(\epsilon) so that if x > T, then the number of m \leq x which have a divisor p-1 > T, with p prime is less than \epsilon x. The paper concludes with several related open questions.
Reviewer:  P.Erdös
Classif.:  * 11B39 Special numbers, etc.
Keywords:  positive asymptotic density; divisibility; Bernoulli number; fractional part; shifted prime

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