On the Diophantine equation $(p^n)^x+(4^mp+1)^y=z^2$ when $p, 4^mp+1$ are prime numbers

Abstract

In this paper, we solve the Diophantine equation $(p^n)^x+(4^mp+1)^y=z^2$ in $\mathbb{N}$ for $p\geq 3$ and $1+4^mp$ 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 $n=3$.