On a class of generalized Fermat equations of signature $(2,2n,3)$

TL;DR

This paper investigates a class of generalized Fermat equations of the form $7x^{2} + y^{2n} = 4z^{3}$, determining solutions for specific small n, formulating criteria for prime p, and verifying solutions computationally for large primes.

Contribution

It provides explicit solutions for small n, develops a Kraus type criterion for primes p, and uses advanced number theory methods to show non-existence of solutions for many primes.

Findings

Solutions explicitly determined for n=2,3,4,5

No solutions for many primes p > 7 verified computationally

Uses symplectic method and quadratic reciprocity to prove non-existence for a positive density of primes

Abstract

We consider the Diophantine equation $7x^{2} + y^{2n} = 4z^{3}$. We determine all solutions to this equation for $n = 2, 3, 4$ and $5$. We formulate a Kraus type criterion for showing that the Diophantine equation $7x^{2} + y^{2p} = 4z^{3}$ has no non-trivial proper integer solutions for specific primes $p > 7$. We computationally verify the criterion for all primes $7 < p < 10^9$, $p \neq 13$. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation $7x^{2} + y^{2p} = 4z^{3}$ has no non-trivial proper solutions for a positive proportion of primes $p$. In the paper \cite{ChDS} we consider the Diophantine equation $x^{2} +7y^{2n} = 4z^{3}$, determining all families of solutions for $n=2$ and $3$, as well as giving a (mostly) conjectural description of the solutions for $n=4$ and primes $n \geq 5$.