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Claim. The equation $x^xy^y=z^z$ has infinitely many solutions in integers $x,y,z>1$, so the answer to Problem 674 is yes. The family is indexed by $n\ge2$ ($n=1$ gives $y=1$): with $a=2^{n+1}$ and $b=2^n-1$,

$$x=2^{a(b-n)}b^{2b}\cdot2^{2n},\qquad y=2^{a(b-n)}b^{2b}\cdot b^2,\qquad z=2^{a(b-n)}b^{2b}\cdot2^{n+1}b.$$

Its member at $n=2$ is $x=2^{12}3^6$, $y=2^83^8$, $z=2^{11}3^7$. The same note proves that no solution has $\gcd(x,y)=1$, so every solution shares a common factor.

Claimant and source. Chao Ko, Note on the Diophantine equation $x^xy^y=z^z$, J. Chinese Math. Soc. (1940), 205–207 [Ko40]. The attribution of the family to Ko rests on Erdős's own report in Erdős 1979, item 10, that Chao Ko found infinitely many solutions, on Mills's 1959 account [Mi59], which the site summarizes as showing that the only solutions with $4xy=z^2$ are Ko's, and on the site's commentary. The page name carries the stand-in date 1940-01-01: the journal issue records no month or day.

Acceptance. The site's curator, Thomas Bloom, labels the problem proved and credits Ko's families in the page's commentary (page last edited 2026-02-01), which is the reviewed evidence; the community database (teorth/erdosproblems, data/problems.yaml) records the problem as proved (Lean). The journal publication in J. Chinese Math. Soc. is the refereed evidence.

Formalization. Bhavik Mehta posted a Lean proof in the site's thread on 2025-12-09, written from the solutions described on the page. The file is retained in Boris Alexeev's lean-proofs repository at the pinned commit above, names Mehta as its formal author, and proves that the solution set $\{(x,y,z): x,y,z>1,\ x^xy^y=z^z\}$ is nonempty, by the $n=2$ member, and infinite, by Ko's parametrized family; the formal-conjectures statement file for the problem names that file on the repository's main branch as its formal proof and tags the problem as solved. This corpus has not built the file or audited its statement, so it is a link and not formalized evidence.

What remains. Erdős asked in 1979 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 found none with $4xy<z^2$ and $\gcd(x,y)<6^{150}$; Schinzel [Sc58] conjectured and Dem'janenko 1975 proved that $x$, $y$ and $z$ share the same prime divisors in every solution; and Uchiyama 1984 showed that each fixed index $xy/z^2<1/4$ admits only finitely many solutions and excluded several families of indices. These results bear on the uniqueness question, not on the claim above.