Generalized Fruit Diophantine equation and Hyperelliptic curves

Abstract

We show the insolvability of the Diophantine equation $a{x}^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1(\mathrm{mod}12)$ and $b=2^{d}a-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we investigate the more general family with $b=2^{d}a-3^r$, where $r$ is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over $\mathbb{Q}$. We conclude by providing some numerical evidence corroborating the main results.

11 pages