##
**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 B_{2 k} are dense in the interval (0,1). Furthermore, for every positive integer k, the set of all m for which B_{2 m} has the same fractional part as B_{2 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag