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erdős #1108

← #1107 · #1109 → (packet.json; erdosproblems.com)

Let $A = {n \in S \sum n! : S \subset N finite} .$ If $k \geq 2$ , then does $A$ contain only finitely many $k$ th powers? Does it contain only finitely many powerful numbers?

Worked, still open.

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

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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

Both questions are **open** (as of early 2026).

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_1108.parts.i : answer(sorry) ↔ ∀ k ≥ 2,
    Set.Finite { a | a ∈ FactorialSums ∧ ∃ m : ℕ, m ^ k = a }

formal-conjectures/1108.lean ↗

oeis

A025494 — Squares which are the sum of factorials of distinct integers (probably finite).1,4,9,25,121,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449 A051761 — Numbers that are simultaneously a sum of factorials of distinct integers and of the form a^b with b >= 2.0,1,4,8,9,25,27,32,121,128,144,729,841,5041,5184,45369,46225,363609,403225,3674889,1401602635449 A115645 — Powerful(1) numbers (A001694) that are sums of distinct factorials.1,8,9,25,27,32,121,128,144,729,841,864,5041,5184,40328,41067,45369,45387,46208,46225,363609,403225,3674889,43954688,6230

status

open

finding.noted · reviewer:will-blair · 1 day

renders the record as of vev_d199cb2e · 1,338 events · hub