On the extensibility of the D(4)-triple {k–2, k+2, 4k} over Gaussian integers

Abdelmejid Bayad, Appolinaire Dossavi-Yovo, Alan Filipin, and Alain Togbé
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 23, 2017, Number 3, Pages 1—26
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Authors and affiliations

Abdelmejid Bayad
Departement de mathématiques, Université d’Evry Val d’Essonne ´
23 Bd. De France, 91037 Evry Cedex, France

Appolinaire Dossavi-Yovo
Institut de Mathematiques et de Sciences Physiques
Porto-Novo, Bénin

Alan Filipin
Faculty of Civil Engineering, University of Zagreb
Fra Andrije Kacica-Milosica 26, 10000 Zagreb, Croatia

Alain Togbé
Mathematics Department, Purdue University North Central
1401 S, U.S. 421, Westville IN 46391 USA

Abstract

In this paper, we prove that if {k – 2, k + 2, 4k}, where k ∈ ℤ[i], k ≠ 0, ± 2, is a D(4)-quadruple in the ring of Gaussian integers, then d = 4k3 − 4k.

Keywords

  • Diophantine m-tuple
  • Pell equation
  • Linear forms in logarithms

AMS Classification

  • 11D09
  • 11D45
  • 11B37
  • 11J86

References

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  2. Bayad, A., Filipin, A. & Togbé, A. (2016) Extension of a parametric family of Diophantine triples in Gaussian integers, Acta Math. Hungar., 148(2), 312–327.
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Cite this paper

APA

Bayad, A., Dossavi-Yovo, A., Filipin, A., & Togbé, A. (2017). On the extensibility of the D(4)-triple {k–2, k+2, 4k} over Gaussian integers, Notes on Number Theory and Discrete Mathematics, 23(3), 1-26.

Chicago

Bayad, Abdelmejid, Appolinaire Dossavi-Yovo, Alan Filipin, and Alain Togbé. “On the extensibility of the D(4)-triple {k–2, k+2, 4k} over Gaussian integers.” Notes on Number Theory and Discrete Mathematics 23, no. 3 (2017): 1-26.

MLA

Bayad, Abdelmejid, Appolinaire Dossavi-Yovo, Alan Filipin, and Alain Togbé. “On the extensibility of the D(4)-triple {k–2, k+2, 4k} over Gaussian integers.” Notes on Number Theory and Discrete Mathematics 23.3 (2017): 1-26. Print.

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