A parametric family of quartic Thue inequalities

Salah Eddine Rihane, Mohand Ouamar Hernane and Alain Togbé
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 1—14
DOI: 10.7546/nntdm.2021.27.4.1-14
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Authors and affiliations

Salah Eddine Rihane
Department of Mathematics and Computer Science
Abdelhafid Boussouf University
Mila 43000, Algeria

Mohand Ouamar Hernane
Université des Sciences et de la Technologie Houari-Boumedièene (USTHB),
Faculté de Mathématiques, Laboratoire d’Algèbre et Théorie des Nombres
BP 32, 16111 Bab-Ezzouar, Alger, Algerie

Alain Togbé
Department of Mathematics, Statistics
Purdue University Northwest
1401 S, U.S. 421, Westville IN 46391, United States

Abstract

Let c\neq 0,-1 be an integer. In this paper, we use the method of Tzanakis to transform the quartic Thue equation x^4 -(c^2+c+4) x^3y +(c^2+c+3) x^2 y^2 +2 xy^3 -y^4 = \mu into systems of Pell equations. Then, we determine all primitive solutions (x,y) with 0<|\mu|\leq |c+1|.

Keywords

  • Parametric Thue equations
  • Method of Tzanakis
  • Continued fraction
  • Linear form in logarithms

2020 Mathematics Subject Classification

  • 11D59
  • 11D09
  • 11D75
  • 11D25
  • 11A55
  • 11J86

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Cite this paper

Rihane, S. E., Hernane, M. O., & Togbé, A. (2021). A parametric family of quartic Thue inequalities. Notes on Number Theory and Discrete Mathematics, 27(4), 1-14, doi: 10.7546/nntdm.2021.27.4.1-14.

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