this post was submitted on 20 Jan 2025
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This person has found an actual solution:
https://mlzeng.com/an-interesting-equation.html
This is apparently the smallest solution.
The derivation of the solution itself requires an understanding of Diophantine equations and elliptical curves, which I will not pretend to understand well enough to summarize.
From the same article, this is a solution when negative integers are allowed:
{{x -> 11, y -> 9, z -> -5}}