Erdős Problem #1050 — Irrationality of a shifted binary Lambert series

Erdős recorded the question in 1988: determine whether the series with denominator $2^n-3$ is irrational.

Known result reported in 2024

A 2024 arXiv paper states that Borwein proved the more general result that $\sum_{n=1}^{\infty}1/(t^n+q)$ is irrational for integer $t\ge2$ and rational $q\ne0,-t^n$ for every $n$. Substituting $t=2$ and $q=-3$ gives the Erdős problem, since $-3\ne0$ and $-3\ne-2^n$ for every $n$; the paper also mentions an alternative proof elsewhere.

Current status (as of September 2026): The sum is reported irrational by Borwein's general theorem, so the problem is claimed solved; this report does not independently verify the theorem.