Problem 251

Erdős Problem #251: declared status 'open'. Formalized: yes. Is$$\sum \frac{p_n}{2^n}$$irrational? (Here $p_n$ is the $n$th prime.) Current best: Erd\H{o}s [Er58b] proved that $\sum \frac{p_n^k}{n!}$ is irrational for every $k\geq 1$. In [Er88c] he further conjectures that $\sum \frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\geq 2$ and $g_n=o(p_n)$ then$$\sum_{n=1}^\infty \frac{p_n}{g_1\cdots g_n}$$is irrational. Prize: no. OEIS: A098990. Tags: irrationality, number theory.