On the Diophantine equation (pⁿ) ˣ+ (4ᵐp+1) ʸ=z² when p, 4ᵐp+1 are prime numbers

In this paper, we solve the Diophantine equation (pⁿ) ˣ+ (4ᵐp+1) ʸ=z² in N for p 3 and 1+4ᵐp are prime integers. Concretely, using the congruent method, we prove that this equation has no non-negative solutions if p>3. For the case p=3, we will show that this equation has no solutions if m>1. Furthermore, in this case, when m=1 using the elliptic curves, we will show that this equation has only solution (x, y, z) = (3, 2, 14) if n=1 and (x, y, z) = (1, 2, 14) if n=3.