Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  080.03305
Autor:  Erdös, Paul
Title:  Sur certaines séries à valeur irrationnelle. (On certain series with irrational values) (In French)
Source:  Enseign. Math., II. Ser. 4, 93-100 (1958).
Review:  Let pn be the nth prime. The author states that he has proved the irrationality of the series sumn = 1oo {pnk \over n!} (k = 1,2,3,...), and he gives the proof for k = 1. It is based on the fact that the numbers pn/n-[pn/n] lie dense in (0,1). More generally, let {qn } be a sequence of integers such that 1 < q1 \leq q2 \leq q3 \leq ···, qn > c n/(log n)k (c > 0, k > 0 constants). Then he shows that

sumn = 1oo {pn \over q1q2··· qn}

is irrational unless qn = qpn+1 for n \geq n0, where q \geq 1 is an integer. This result can be further improved.
Reviewer:  K.Mahler
Classif.:  * 11J72 Irrationality
Index Words:  Number Theory

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