Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 131.04902
Autor: Erdös, Pál; Straus, E.G.
Title: On the irrationality of certain Ahmes series (In English)
Source: J. Indian Math. Soc., n. Ser. 27, 129-133 (1963).
Review: The authors prove that if n1,n2,... is an increasing sequence of positive integers such that (i) limsup n2k /nk+1 \leq 1 and (ii) the sequence {Nk/nk+1} is bounded, Nk denoting the least common multiple of n1,n2,...,nk, then sum 1/nk is rational if and only if nk+1 = nk2-nk+1 for all k \geq k0. The authors proceed to examine how far conditions (i) and (ii) are necessary and prove, in particular, that (ii) can be replaced by limsup (Nk/nk+1) {n2k+2/nk+2-1} \leq 0. Finally three specific examples of irrational Ahmes series sum 1/nk are given, one being due to S.W.Golomb (Zbl 115.04501).
Reviewer: A.Baker
Classif.: * 11J72 Irrationality
Index Words: number theory
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