Discussion on question by Rodrigo Pizarro: Prove that $x^x+y^y=z^z$ doesn't have integer solutions

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04:52

Q: Prove that $x^x+y^y=z^z$ doesn't have integer solutions

Prove that $x^x+y^y=z^z$ doesn't have integer solutions To be honest, I don't see any way to start this problem, I tried for hours but it's not as easy as I thought. Any hints? As you can see in the comments, there is a solution for natural numbers, the problem is when the set is extended to i...

Robert Israel

I presume you're not allowing $0^0$?

Macavity

Positive integers, one presumes? Otherwise it may get ugly fast.

Rodrigo Pizarro

Isn't duplicate because with primes the problem is a lot easier

Macavity

The answer there covers all natural numbers too, not just primes.

Rodrigo Pizarro

I know it's related but still doesn't answer the question for integers

Jacob Bond

04:52

@MartinR This question says integers. The ability to subtract one of the terms opens up new possibilities.

Rodrigo Pizarro

Yeah, thats what they don't understand, but they marked this as duplicate and i don't want to fight, so, well...

Jacob Bond

If one of them is negative, do you want that exponent to still be positive?

I'm not sure if you ran any computer tests already, but there are no solutions with $1 \leq \lvert x \rvert, \lvert y \rvert, \lvert z \rvert \leq 100$.

Rodrigo Pizarro

I didn't, this is a math contest problem of my country, not prizes or something in between, the problem was from some years ago.

Mircea

You can search for Mihai Cipu on arxiv. He has several papers regarding exponential Diophantine equations; maybe you could find some hints.

Erick Wong

The problem is an interesting problem, but the question lacks all sorts of context that has been raised in the comments. Please edit.

For the negative case it seems like one could make a good deal of headway considering the factorizations of denominators (possibly combined with size considerations).

Rodrigo Pizarro

04:52

If i found a "possible" solution, should I put it in my post, or just answer my own question below?