TW-TST 1993 Problem 3

Find all $x,y,z\in\mathbb{N}_{0}$ such that $7^{x} + 1 = 3^{y} + 5^{z}$ .

Alternative formulation: Solve the equation $1+7^{x}=3^{y}+5^{z}$ in nonnegative integers $x$ , $y$ , $z$ .