Are there infinitely many solutions to $\frac{1}{p _{1}} + \dots + \frac{1}{p _{k}} = 1 - \frac{1}{m},$ where $m \geq 2$ is an integer and $p_{1} < \dots < p_{k}$ are distinct primes?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=none) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'none'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

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

Let $ \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m,\qquad m\ge2,\quad p_1<\cdots<p_k\ \text{distinct primes}. $

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_313 : answer(sorry) ↔ erdos313Solutions.Infinite

open