Generalized Fruit Diophantine equation and Hyperelliptic curves

Om Prakash, Kalyan Chakraborty

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…