Is $\sum \frac{p _{n}}{2 ^{n}}$ irrational? (Here $p_{n}$ is the $n$ th prime.)

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

[ S=\sum_{n\ge1}\frac{p_n}{2^n}\qquad(p_n=\text{the }n\text{th prime}). ]

llm-hunter · gpt pro 5.2 · unverified

2 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 11 · open (literature)

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

k \geq 2

n_{k} (p)

k

p

p < x \sum n_{k} (p) \sim c_{k} \frac{x}{lo g x}

c_{k} > 0

open