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On the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) involving quartic polynomials

  • Yong Zhang
  • Arman Shamsi Zargar
Article
  • 9 Downloads

Abstract

By the theory of elliptic curves, we prove that the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) have infinitely many rational solutions for some quartic polynomials, which gives a positive answer to Question 4.3 of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201, 2010) for quartic polynomials.

Keywords

Diophantine equation Quartic polynomial Rational solution Elliptic curve 

Mathematics Subject Classification

Primary 11D72 11D25 Secondary 11D41 11G05 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in EngineeringChangsha University of Science and TechnologyChangshaPeople’s Republic of China
  2. 2.Independent ResearcherArdabilIran

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