OPEN
This is open, and cannot be resolved with a finite computation.
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?
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.
View the LaTeX source
This page was last edited 28 September 2025.
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #247, https://www.erdosproblems.com/247, accessed 2025-11-16
All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic.
