Generalized Fruit Diophantine equation over number fields

Abstract

Let $K$ be a number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the solutions of the generalized fruit Diophantine equation $ax^d-y^2-z^2 +xyz-c=0$ over $K$, where $d \geq 3$ is an integer and $a,c\in \mathcal{O}_K\setminus \{0\}$. Subsequently, we provide explicit values of square-free integers $t$ such that the equation $ax^d-y^2-z^2 +xyz-c=0$ has no solution $(x_0, y_0, z_0) \in \mathcal{O}_{\mathbb{Q}(\sqrt{t})}^3$ with $2 | x_0$, and demonstrate that the set of all such square-free integers $t$ with $t \geq 2$ has density exactly $\frac{1}{6}$. As an application, we construct infinitely many elliptic curves $E$ defined over number fields $K$ having no integral point $(x_0,y_0) \in \mathcal{O}_K^2$ with $2|x_0$.