On the Solutions of the Diophantine Equation pˣ + (p+4k) ʸ=z² for Prime Pairs p and p+4k

In this paper, we solve the Diophantine equation px + (p + 4k) y = z2 in N0 for prime pairs (p, p+ 4k). First, we consider cousin primes p and p+ 4. Then we extend the study to solving px + (p + 4) y = z 2n, where n ∈ N\1. Furthermore, we solve the equation px + (p + 4k) y = z2 for k ≥ 2. As a result, we show that this equation has a unique solution (p, p + 4k, x, y, z) = (3, 11, 5, 2, 122) whenever x > 1 and y > 1. Finally, we show the finiteness of number of solutions in N.