Weighted Fruit Diophantine Equations and Hyperelliptic Curves

Abstract

We study the weighted fruit Diophantine equation $a{x}^d-c(m^2{y}^2+n^2{z}^2)+xyz-b=0$, generalising previous work by Majumdar--Sury, Vaishya--Sharma, and Prakash--Chakraborty. Subject to specific hypotheses on the parameters, our main result shows that for any prime $l\equiv 3(\mathrm{mod}4)$ and $b=a(2cmn{)}^d-l{c}^s{t}^{2q}$, the above equation has no integer solutions except for certain residue classes of $x$ modulo $4l$. An analogous result also holds when $l$ is replaced by an odd power of $l$ in the definition of $b$. We prove some insolvability results for $l=-1$. By applying the main result to the small values of $l$, such as $l\in \{3,7,11,19\}$, we explicitly determine the exceptional residue classes outside of which the equation has no solutions. In particular, for $l=3$, this yields complete insolvability, and weakening these hypotheses still yields non-existence results, though with specific coprimality restrictions on any possible solutions. We also consider a more general variant of the above Diophantine equation and provide some insolvability results. Subsequently, we establish bounds for the positive solutions of the aforementioned equation. Finally, by associating a family of hyperelliptic curves with the equation under consideration and applying Grant's analogue of the Nagell--Lutz theorem, we translate these insolvability results into results about their rational torsion points.