I'm working on a problem where we must show that there are infinitely many integer triples $x,y,z$ such that $x^2 + y^2 + z^2$ is divisible by $(x + y +z)$ and $x,y,z$ are pairwise coprime. Also $x,y,z$ are distinct.
I can see that either all $x,y,z$ are odd or that one of them is even but besides that haven't made much progress (I can't see any nice factorization of $x^2 + y^2 + z^2$ nor have I been able to find an easy example to get a foothold).
Broadly I'm thinking that my two major approaches are proof by contradiction that the number of triples can't be finite, or trying to find some construction to generate the triples.
I imagine finding a construction would be more fruitful.
I would appreciate both a solution but also advice on how to approach this type of problem