## 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

number theory

Zbl 0115.04501