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 $7<p<10^9$, $p\ne 13$. 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 $n\ge 5$.

31 pages