Integral solutions of certain Diophantine equation in quadratic fields

TL;DR

This paper investigates the solvability of a specific Diophantine system within the rings of integers of quadratic fields, establishing non-solvability for most cases except a few specific fields.

Contribution

It proves that the Diophantine system has solutions only in the rings of integers of four particular quadratic fields, providing a classification of solvability.

Findings

Solutions exist only in rings of integers of $d= -7, -1, 17, 101$

For all other quadratic fields, the system is not solvable

The result classifies solvability across quadratic fields

Abstract

Let $K= \mathbf{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ be its ring of integers. We study the solvability of the Diophantine equation $r + s + t = rst = 2$ in $\mathcal{O}_{K}$. We prove that except for $d= -7, -1, 17$ and $101$ this system is not solvable in the ring of integers of other quadratic fields.