Integral solutions of certain Diophantine equation in 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.

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