Proceedings of the Japan Academy, Series A, Mathematical Sciences

Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields

Ljerka Jukić Matić

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Abstract

In this paper we derive some elements of the rings of integers in the cubic fields of the form $\mathbf{Q}(\sqrt[3]{d})$, where $d$ is even, which cannot be written as a difference of two squares in the considered ring. We show that corresponding Diophantine quadruples do not exist for such elements, what supports the hypothesis mainly proved for the ring of integers and for certain quadratic fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 10 (2012), 163-167.

Dates
First available in Project Euclid: 6 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.pja/1354802414

Digital Object Identifier
doi:10.3792/pjaa.88.163

Mathematical Reviews number (MathSciNet)
MR3004232

Zentralblatt MATH identifier
1284.11057

Subjects
Primary: 11D09: Quadratic and bilinear equations 11R16: Cubic and quartic extensions
Secondary: 11D79: Congruences in many variables

Keywords
Diophantine quadruples cubic fields

Citation

Jukić Matić, Ljerka. Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 10, 163--167. doi:10.3792/pjaa.88.163. https://projecteuclid.org/euclid.pja/1354802414


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