Problem 1108

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Statement. Let

$$A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}.$$

If $k\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?

Status. Open.

Source. erdosproblems.com/1108, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1108, https://www.erdosproblems.com/1108.

References.

• [BrEr91] Brindza, B. and Erdős, P., On some Diophantine problems involving powers and factorials. J. Austral. Math. Soc. Ser. A 51 (1991), no. 1, 1-7.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• brindza_1991_diophantine_problems_involving_powers_factorials
• brindza_1991_diophantine_problems_involving_powers_factorials / theorem_1