Prime numbers only

z must be odd so: $\ 2^y +1 = z$ Now the left hand side must be a fermat prime. i.e. $\ y = 2^n$ Clearly y is only prime for n = 1 which produces $\ (x,y,z) = (2,2,5)$ as a unique triple

Approach without knowledge of Fermat Prime:

We note that $z$ must be odd, because if $z$ is even then $x^{y}$ must be 1. Which is impossible for prime $x,y$ .

If $z$ is odd, then $x^{y}$ must be even, because even + odd = odd.

If $x^{y}$ is even, then $x$ must be an even number, because an odd number raised to any power will still be an odd number.

The only even prime number is 2, hence $x=2$ .

$y$ must also be even because else $x^{y} + 1$ is factorizable, making $z$ composite.

Hence $y$ is also 2.

Fill it in and we have $(x,y,z)=(2,2,5)$ .