The board / number theory / #674
Erdős Problem #674
PROVED LEANdatabase updated 2025-12-08
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The set of integer solutions to \(x^x y^y = z^z\) with \(x, y, z > 1\).
Are there any integer solutions to \(x^xy^y=z^z\) with \(x,y,z>1\)? Ko [Ko40] proved there are none if \((x,y)=1\), but there are in fact infinitely many solutions in general - for example, \(x=2^{12}3^6\), \(y = 2^83^8\), and \(z = 2^{11}3^7\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/674.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/674
Attempts
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