There is the so-called “Proteus theorem or heteroscaling theorem,” which was probably first voiced (as a separate statement) by Ib Petrov (Ib is a real name, he was surprised himself) in his article of the same name. Petrov Ib. "Proteus' Theorem or Heteroscaling Theorem", self-published, 2024. — 6 с. (publication in Russian). Actually the theorem goes like this: [U]Theorem[/U]: For integers [B]a[/B], [B]b[/B] and [B]c[/B], where [B]c > 1[/B], [B]a > 1[/B] and [B]b > 1[/B], the equations [B]a^b+b^a=c^a[/B] and [B]a^b+b^a=c^b[/B] do not have complete solutions. In essence, this is a special case of Diophantine equations. And it seems that the proof given by the author is understandable (I would say even on an intuitive level), but somehow it seems to me not entirely formal, in terms of its sufficiency. The topic is interesting, since here there is a relationship between the exponent and the base, as well as the “rate” of growth of functions on the right and left sides of the equation. The author applies proof by contradiction, based on the fact that the function ([B]a^b+b^a[/B]) has polynomial growth, and [B]c^a[/B] has exponential growth. It seems to me that this is not a formal (sufficient) proof, for example, for very large natural numbers. Although in essence these equations are a special case of Fermat’s Last Theorem - if you think about it...