Generalized Fruit Diophantine equation and Hyperelliptic curves

TL;DR

This paper proves the insolubility of a specific class of Diophantine equations under certain conditions and explores related hyperelliptic curves, revealing an infinite family with trivial torsion over rationals.

Contribution

It establishes new insolubility results for a generalized Diophantine equation and identifies an infinite family of hyperelliptic curves with trivial torsion.

Findings

Insolvability of the equation for specified parameters.

Existence of an infinite family of hyperelliptic curves with trivial torsion.

Numerical evidence supporting theoretical results.

Abstract

We show the insolvability of the Diophantine equation $ax^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1 \pmod {12}$ and $b=2^da-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we investigate the more general family with $b=2^da-3^r$, 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.