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 \in \mathbb{Z}[i], k \ne 0, \pm 2, is a D(4)-quadruple in the ring of Gaussian integers, then d = 4k^3 - 4k.

Keywords

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

AMS Classification

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

References

  1. Baker, A. & Wüstholz, G. (1993) Logarithmic forms and group varieties, J. Reine Angew. Math. 442, 19–62.
  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.
  3. Bennett, M. A. (1998) On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 , 173–199.
  4. Cipu, M. & Trudgian, T. (2016) Searching for Diophantine quintuples, Acta Arith., 173, 365-382.
  5. Dujella, A. Diophantine m-tuples, http://web.math.pmf.unizg.hr/~duje/dtuples.html.
  6. Dujella, A. (2001) An absolute bound for the size of Diophantine m-tuples, J. Number Theory, 89, 126–150.
  7. Dujella, A. (2004) There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566, 183–214.
  8. Franusic, Z. (2008) On the extensibility of Diophantine triples \{k - 1, k + 1, 4k \} for Gaussian integers, Glas. Mat. Ser. III 43, 265–291.
  9. Fujita, Y. (2006) The unique representation d = 4k(k^2 - 1) in D(4)-quadruples \{ k - 2, k + 2, 4k, d \}, Math. Commun. 11, 69–81.
  10. Jadrijevic, B. & Ziegler, V. (2006) A system of relative Pellian equations and related family of relative Thue equations, Int. J. Number Theory 2, 569–590.

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Cite this paper

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.

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