I want to show that the equation
$x^2$ + $y^2$ + $z^2$ = $2xyz$ has no non-trivial solutions using infinite descent.
This question has been solved with infinite descent by showing that $x,y,z$ are all even and we can infinitely find smaller solutions: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
Can it be shown instead by assuming you have the least positive solution and then finding a smaller solution.
I let ($a,b,c$) be the least positive solution with gcd(a,b,c) = 1, and I want to find a smaller solution.
I have that exactly one of $a,b,c$ must be even because if two were even, then the left hand side would be odd while the right hand side is even. And if all three are even, then gcd($a,b,c$) $\ge$ 2.
So we can assume $a$ is even. I've tried considering the equation mod $a,b,c,2$ but couldn't seem to find a way to construct a smaller solution.