On some generalized Fermat equations of the form x2+y2n=zp$x^2+y^{2n} = z^p$

The primary aim of this paper is to study the generalized Fermat equation \[ x^2+y^{2n} = z^{3p} \] in coprime integers $x$, $y$, and $z$, where $n \geq 2$ $p$ a fixed prime. Using modularity results over totally real fields explicit computation Hilbert cuspidal eigenforms, we provide complete resolution case $p=7$, obtain an asymptotic result for $p$. Additionally, using similar techniques, solve second equation, namely $x^{2\ell}+y^{2m} z^{17}$, primes $\ell,m \ne 5$.