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Claim. The Theorem on p. 222 of P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222--226, received 13 December 1965, states that if the integers $n_1<n_2<\cdots$ are pairwise coprime and $\sum_i1/n_i<\infty$, then

$$\sum_{i\ge1}\frac{1}{t^{n_i}-1}$$

is irrational for every integer $t\ge2$. At $t=2$ every such set $A=\{n_i\}$ is an instance of Problem 257, answered yes. The proof writes the sum as $\sum_mV^*(m)/t^m$, equation (3), where $V^*(m)$ counts the $n_i$ dividing $m$, and shows that the base-$t$ expansion of this value is infinite but contains arbitrarily long blocks of zeros: $k$ simultaneous congruences, equation (4), choose $y$ with $V^*(y+i)=t^i$ for $i=1,\ldots,k$, which pairwise coprimality makes possible, and the estimates (5) to (14) bound the digits that follow. Erdős writes that more complicated arguments show the coprimality condition to be superfluous, giving no details, that the convergence condition could be replaced by a weaker one, and that he expects the series to be irrational whenever $n_{k+1}-n_k\to\infty$ and perhaps whenever $n_k/k\to\infty$; on p. 226 he adds that without coprimality the proof needs the fact that a number whose multiples $t^n\alpha$ have infinitely many distinct fractional parts is irrational, and that he cannot handle the case in which the $n_i$ are all the primes. That case is Problem 69, settled by Tao and Teräväinen; see their claim page. The source card is erdos_1969_irrationality_certain_series.

Covers. Every infinite pairwise coprime $A$ with $\sum_{a\in A}1/a<\infty$, for the base $2$ of the question and for every integer base $t\ge2$. Not covered: supports in which some pair shares a factor, and supports with divergent reciprocal sum. Erdős's statement that coprimality is superfluous is made without details, so the sets it would add are not covered here; Cook's pending claim writes out an argument for that extension.

Acceptance. Refereed: the Mathematics Student, volume 36 (1968), pp. 222--226, received 13 December 1965 as the paper's header prints. The site labels the problem OPEN, and its remark crediting the paper with this class is commentary on an open problem, not acceptance, so no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.