FormalConjectures.ErdosProblems.«674»

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import FormalConjecturesUtil

Erdős Problem 674

References:

erdosproblems.com/674

[Ko40] Ko, Chao, Note on the Diophantine equation $x^xy^y=z^z$. J. Chinese Math. Soc. (1940), 31-39.

namespace Erdos674

The set of integer solutions to $x^x y^y = z^z$ with $x, y, z > 1$.

def solutionSet : Set (ℕ × ℕ × ℕ) :=
  {(x, y, z) | 1 < x ∧ 1 < y ∧ 1 < z ∧ x ^ x * y ^ y = z ^ z}

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 $z = 2^{11}3^7$.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos674.lean"]
theorem erdos_674 : answer(True) ↔ solutionSet.Nonempty := by⊢ True ↔ solutionSet.Nonempty
  sorryAll goals completed! 🐙

There are in fact infinitely many integer solutions to $x^xy^y=z^z$ with $x,y,z>1$.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos674.lean"]
theorem erdos_674.variants.infinite : solutionSet.Infinite := by⊢ solutionSet.Infinite
  sorryAll goals completed! 🐙

end Erdos674