On the Diophantine Equation pˣ+ (p+1) ʸ=z², where p is Prime

In this paper, we study all non-negative integer solutions of the Diophantine equation pˣ+ (p+1) ʸ=z², where p is a prime number. The research results showed that 1) If y=0, then the equation has only two solutions, namely (p, x, z) ∈ (2, 3, 3), (3, 1, 2), 2) If y=1, then the equation has only solutions in the form of (p, x, z) = (2, 0, 2) or (p, x, z) = (4n²+4n-1, 0, 2n+1), where n is a positive integer, 3) If y=2, then the equation has only two solutions, namely (p, x, z) ∈ (2, 4, 5), (3, 2, 5), 4) If p>3 and y is even, then the equation has no solution, and 5) If p=5, then the equation has only one solution. That is (x, y, z) = (4, 3, 29).