Erdős Problem #1108 — open problem in number theory

Erdős Problem #1108

For each k ≥ 2, does the set A = Σ_n∈ Sn! : S⊂ ℕ finite of all finite sums of distinct factorials contain only finitely many k-th powers?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-1108,
  title        = {Erdős Problem #1108},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1108}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_1108.parts.i. For each $k \geq 2$ , does the set $A = {\sum_{n \in S} n! : S \subset N finite}$ of all finite sums of distinct factorials contain only finitely many $k$ -th powers?

erdos_1108.parts.ii. Does the set $A = {\sum_{n \in S} n! : S \subset N finite}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1108» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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

theorem erdos_1108.parts.ii :
     answer(sorry) ↔ {a ∈ FactorialSums | IsPowerful a}.Finite

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1108. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1108

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.