Problem 674

Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$ ?

Number theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN) (page last edited 1 February 2026); the accepted claim on Ko's infinite family of solutions settles it.

De75b: Demʹjanenko, V. A., On a conjecture of A. Schinzel. Izv. Vysš. Učebn. Zaved. Matematika (1975), 39-45.
Er79: Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
Ko40: Ko, Chao, Note on the Diophantine equation $x^xy^y=z^z$ . J. Chinese Math. Soc. (1940), 205-207.
Mi59: W. H. Mills, An unsolved Diophantine equation. Rep. Inst. in the theory of numbers, University of Colorado (1959), 258-268.
Sc58: Schinzel, A., Sur un problème de P. Erdős. Colloq. Math. (1958), 198-204.
Uc84: Uchiyama, S., On the Diophantine equation $x^x y^y=z^z$ . Trudy Mat. Inst. Steklov. (1984), 237-243.

Problem 674

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Proof claims

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