Is $n \geq 2 \sum \frac{1}{n ! - 1}$ irrational?

Worked, still open.

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

machinery: irrationality-measure,factorial-series,transcendence,linear-forms-in-values,Liouville-approximation,prime-distribution

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

This exact question is listed as an open Erdős problem (Erdős Problem #68): it asks whether [ \sum_{n\ge 2}\frac{1}{n!-1} ] is irrational, and it is still marked **OPEN**. ([Erdős Problems][1])

candidate solution ↗

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

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_68 :
    answer(sorry) ↔ Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ))

formal-conjectures/68.lean ↗

oeis

A331373 — Decimal expansion of Sum_{k>=2} 1/(k! - 1).1,2,5,3,4,9,8,7,5,5,6,9,9,9,5,3,4,7,1,6,4,3,3,6,0,9,3,7,9,0,5,7,9,8,9,4,0,3,6,9,2,3,2,2,0,8,3,3,2,0,1,3,4,1,7,0,6,3,8,3,

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open

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