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 $n=3$.