Erdős Problem #257

declared status 'open'. Formalized: yes. Let $A\subseteq \mathbb{N}$ be an infinite set. Is$$\sum_{n\in A}\frac{1}{2^n-1}$$irrational? Current best: There is nothing special about $2$ here, and this sum is likely irrational with $2$ replaced by any integer $t\geq 2$. This was disproved by Kova\v{c} and Tao [KoTa24], and in the comments Kova\v{c} has sketched a proof that there exists some choice of $t_n$ with $1\leq t_n\leq 6$ such that this sum is rational. Prize: no. Tags: irrationality.

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