Problem 399

Is it true that there are no solutions to

n! = x^k\pm y^k

with $x,y,n\in \mathbb{N}$ , with $xy>1$ and $k>2$ ?

Source: erdosproblems.com/399

Number theory
Factorials

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The site labels the problem DISPROVED (LEAN) (page last edited 30 September 2025) and credits Jonas Barfield with the solution $10!=48^4-36^4$ , recorded on its claim page (Barfield, 2025); the Lean qualifier refers to the formal-conjectures file, which checks the witness by decide and which this corpus has not built. There is no refereed write-up. The standing in the frontmatter derives from the claim pages.

Br32: Breusch, Robert, Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen $x$ und 2 $x$ stets Primzahlen liegen. Math. Z. (1932), 505-526.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 77. Library home: Erdős and Graham 1980.
ErOb37: Erdős, P. and Obláth, R., Über diophantische Gleichungen der Form $n!=x^p\pm y^p$ und $n!\pm m!=x^p$ . Acta Litt. ac Sci. Reg. Univ. Hung. Fr.-Jos., Sect. Sci. Math. 8 (1937), 241-255. Library home: Erdős and Obláth 1937.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section D25 "Equations involving factorial $n$ ", printed p. 301: "Erdős & Obláth dealt with the equation $n!=x^p\pm y^p$ with $x\perp y$ and $p>2$ ". Library home: Guy 2004.
PoSh73: Pollack, Richard M. and Shapiro, Harold N., The next to last case of a factorial diophantine equation. Comm. Pure Appl. Math. 26 (1973), 313-325.

Problem 399

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