Problem 247

Erdős Problem #247: declared status 'open'. Formalized: yes. 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? Current best: Erd\H{o}s [Er75c] proved the answer is yes under the stronger condition that $\limsup n_k/k^t=\infty$ for all $t\geq 1$. Prize: no. Tags: irrationality, number theory.