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

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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.

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References

  1. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Cohen, Number Theory, Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, vol. 239 (Springer, New York, 2007)

    Google Scholar 

  3. B. He, A. Togbé, M. Ulas, On the Diophantine equation \(z^2=f(x)^2\pm f(y)^2\), II. Bull. Aust. Math. Soc. 82(2), 187–204 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. L.J. Mordell, Diophantine Equations, Pure and Applied Mathematics, vol. 30 (Academic, London, 1969)

    Google Scholar 

  5. J.H. Silverman, J. Tate, Rational Points on Elliptic Curves (Springer, New York, 1992)

    Book  MATH  Google Scholar 

  6. Sz Tengely, M. Ulas, On certain Diophantine equations of the form \(z^2=f(x)^2\pm g(y)^2\). J. Number Theory 174, 239–257 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Ulas, A. Togbé, On the Diophantine equation \(z^2=f(x)^2\pm f(y)^2\). Publ. Math. Debrecen 76(1–2), 183–201 (2010)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, 410114, People’s Republic of China

    Yong Zhang

  2. Independent Researcher, Ardabil, Iran

    Arman Shamsi Zargar

Authors
  1. Yong Zhang
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  2. Arman Shamsi Zargar
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Corresponding author

Correspondence to Yong Zhang.

Additional information

This research was supported by the National Natural Science Foundation of China (Grant No. 11501052).

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Zhang, Y., Zargar, A.S. On the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) involving quartic polynomials. Period Math Hung 79, 25–31 (2019). https://doi.org/10.1007/s10998-018-0259-7

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  • DOI: https://doi.org/10.1007/s10998-018-0259-7

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