another counter-example

Let's look at the familiar equation 3^2 + 4^2 = 5^2

What happens to 3^3 + 4^3 = 73 ? The cube root of 73 is 4.17 (approximately)

What happens to 3^4 + 4^4 = 291 ? The 4th root is 4.13 (approximately.

The progression of the higher roots is toward the number 4.
5, 4.27, 4.13 As the root gets higher in number, the root gets closer to 4. While the square root may be greater than y + 1, eventually the distance between z and y is less than 1. this means that there is a finite number of possible solutions for z, when x,y are fixed. So there are only a finite number of steps to consider for each z.