Explicit solutions and non-solutions for the Diophantine equation $p^x+q^{2y}=z^{2n}$ involving primes $p\not\equiv q(\mathrm{mod}4)$

Over the past decade, significant research has been conducted on the equation $a^x+b^y=z^2$ under various conditions imposed on $a$ and $b$ or on $x$ and $y$. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where $x$ or $y$ is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form $p^x+q^{2y}=z^{2n}$ for two distinct primes $p$ and $q$, and to present explicit forms of their solutions $(p,x,q,y,z,n)$. Recent studies on the exponential Diophantine equation $p^x+q^y=z^2$, where $p$ and $q$ are primes, have addressed cases where $p=2$ or $p\equiv q(\mathrm{mod}4)$. In this paper, we address the case where $p\not\equiv q(\mathrm{mod}4)$ and $y$ is even. In addition, we explore special cases where $z$ is the prime and provide the complete set of solutions for $p^x+q^{2y}=z^{2n}$. We also show that the equation has no solution when $\{2,3\}⊈\{p,q,z\}$. In other words, we provide almost explicit solutions to $p^x+q^y=z^{2n}$ except for the case where both $x$ and $y$ are odd.