On some Generalized Fermat Equations of the form $x^2+y^{2n}=z^p$

Abstract

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 $n\ge 2$ and $p$ is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a complete resolution of this equation in the case $p=7$, and obtain an asymptotic result for fixed $p$. Additionally, using similar techniques, we solve a second equation, namely $x^{2\ell }+y^{2m}=z^{17}$, for primes $\ell ,m\ne 5$.

To appear in Mathematika. Correction to the proof of Lemma 7.2 and other minor revisions

References in corpus (2)

• Fermat's Last Theorem and modular curves over real quadratic fields
• Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

Cited by in corpus (1)

• Generalised Fermat equation: a survey of solved cases