Bernoulli numbers and exact covering systems. (English) Zbl 0776.11008

Beebee, John

Am. Math. Mon. 99, No. 10, 946-948 (1992).

It is proved that a set of arithmetic progressions \(A=\bigl\{b_ i\pmod{a_ j};\;1\leq j\leq n\bigr\}\) is an exact covering system with \(b_ 1=0\) and \(0\leq b_ j< a_ j\) if and only if the reciprocals of the \(a_ j\)’s sum to 1 and the \(a_ j\), \(b_ j\) generate an interesting recurrence for Bernoulli numbers. The recurrences thus obtained generalize the one found by E. Deeba and D. Rodriguez [Am. Math. Mon. 98, 423-426 (1991; Zbl 0743.11012)].

Reviewer: Št.Znám (Bratislava)

MSC:

11B68	Bernoulli and Euler numbers and polynomials
11A07	Congruences; primitive roots; residue systems
11B25	Arithmetic progressions

Keywords:

arithmetic progressions; exact covering system; recurrence; Bernoulli numbers

Citations:

Zbl 0743.11012