Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 674

claims/: The 1 claim page of Problem 674, one per claimant's result; the problem's standing derives from them.

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

Status. 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.

Source. erdosproblems.com/674, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #674, https://www.erdosproblems.com/674.

References.

[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.

Formalization. Statement in formal-conjectures.

Current assessment

The answer is yes. Ko [Ko40] found infinitely many solutions in integers $x,y,z>1$ , the smallest member of his family being $x=2^{12}3^6$ , $y=2^83^8$ , $z=2^{11}3^7$ , and proved that no solution has $\gcd(x,y)=1$ ; the result is the accepted claim on its claim page, which also links a Lean proof of the family that this corpus has not built. Erdős asked in [Er79] whether Ko's families are the only solutions; that question is open and is not the problem's question. Mills [Mi59] excluded solutions with $4xy>z^2$ and showed that Ko's are the only ones with $4xy=z^2$ ; Dem'janenko [De75b] proved Schinzel's conjecture [Sc58] that $x$ , $y$ and $z$ share the same prime divisors in every solution; and Uchiyama [Uc84] showed that each fixed index $xy/z^2<1/4$ admits only finitely many solutions. These results bear on the uniqueness question, not on the problem's.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (13)

Ko's infinite family of solutions problems/diophantine_problems/E0674/claims Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/demjanenko_1975_conjecture Lemma 1 (p. 40): a solution of x^x y^y = z^z of the shape (4) has n > 1 Lemma 2 (pp. 43-44): a solution of x^x y^y = z^z whose terms do not share their prime divisors has the form (14)Main theorem (p. 39): in every solution of x^x y^y = z^z in natural numbers different from 1, x, y and z have the same prime divisors diophantine_problems/uchiyama_1984_diophantine_equation Theorem 3 (p. 238): each fixed index Q < 1/4 admits at most finitely many non-trivial solutions of x^x y^y = z^z, all effectively determinable Theorem 4 (p. 238): x^x y^y = z^z has no non-trivial solution of index Q = (1-k^2)/4 with k rational, 0 < k < 1 Theorem 5 (p. 238): no non-trivial solution of x^x y^y = z^z has index Q = L/R in lowest terms with Q <= (X+1)/(X+2)^2 with X = [log R/log 2], nor Q < 1/4 with 1 <= L <= 5 number_theory/erdos_1979_unconventional_problems_number_theory_math_mag

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