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On the irrationality of certain Ahmes series. (English) Zbl 0131.04902

The authors prove that if \(n_1,n_2,...\) is an increasing sequence of positive integers such that (i) \(\limsup n^2_k /n_{k+1} \leq 1\) and (ii) the sequence \(\{N_k/n_{k+1}\}\) is bounded, \(N_k\) denoting the least common multiple of \(n_1,n_2,...,n_k\), then \(\sum 1/n_k\) is rational if and only if \(n_{k+1} = n_k^2-n_k+1\) for all \(k \geq k_0\). The authors proceed to examine how far conditions (i) and (ii) are necessary and prove, in particular, that (ii) can be replaced by \[ \limsup (N_k/n_{k+1}) \{n^2_{k+2}/n_{k+2}-1\} \leq 0. \] Finally three specific examples of irrational Ahmes series \(\sum 1/n_k\) are given, one being due to S.W.Golomb (Zbl 0115.04501).
Reviewer: A.Baker

MSC:

11J72 Irrationality; linear independence over a field

Keywords:

number theory

Citations:

Zbl 0115.04501
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