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

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 $n \geq 5$.