Erdős Problem 251

Reference: erdosproblems.com/251

theorem Erdos251.erdos_251 : Irrational (∑' (n : ℕ), ↑(Nat.nth Nat.Prime n) / 2 ^ n) ↔ sorry

Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).