Original scientific paper

Some diophantine quadruples in the ring Z[√-2]

Fadwa S. Abu Muriefah
A. Al-Rashed

Abstract

A complex diophantine quadruple with the property D\,$(z)$, where $z \in $ Z$[\sqrt {-2}]$, is a subset of Z$[\sqrt {-2}]$ of four elements such that the product of its any two distinct elements increased by $z$ is a perfect square in Z$[\sqrt{-2}]$. In the present paper we prove that if $b$ is an odd integer, then there does not exist a diophantine quadruple with the property D$(a + b\sqrt{-2})$. For $z=a+b\sqrt{-2}$, where $b$ is even, we prove that there exist at least two distinct complex diophantine quadruples if $a$ and $b$ satisfy some congruence conditions.

Keywords

quadratic field; diophantine equation

Hrčak ID:

709

URI

https://hrcak.srce.hr/709

Publication date:

26.6.2004.

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