Problem Statement
Let $a_1<a_2<\cdots$ be a sequence of integers such that\[\limsup \frac{a_n}{n}=\infty.\]Is\[\sum_{n=1}^\infty \frac{1}{2^{a_n}}\]transcendental?
Categories:
Number Theory Irrationality
Progress
Erdős [Er75c] proved the answer is yes under the stronger condition that $\limsup n_k/k^t=\infty$ for all $t\geq 1$.Erdős [Er88c] says 'many of these problems seem hopeless at present, but perhaps one can prove that if $a_n>cn^2$ then $\sum_{n=1}^\infty \frac{1}{2^{a_n}}$ is not the root of any quadratic polynomial'.
This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/247 | Last verified: January 14, 2026