Take any pythagorean triplet $(a,b,c)$, we know, by the definition that: $$a^2 + b^2 = c^2$$
But take $$a^x + b^x = c^x$$
Is $x=2$ the only possible solution $\in \Bbb R$ in this case? How can this be concluded?
I conjecture that $2$ is the only solution but I am not sure how to conclusively state this fact.