On a class of quartic Diophantine equations of at least five variables

Hamid Reza Abdolmalki and Farzali Izadi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 1—9
DOI: 10.7546/nntdm.2018.24.3.1-9
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Authors and affiliations

Hamid Reza Abdolmalki
Department of Mathematics, Faculty of Science,
Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran

Farzali Izadi
Department of Mathematics, Faculty of Science,
Urmia University, Urmia 165-57153, Iran

Abstract

In this paper, elliptic curves theory is used for solving the quartic Diophantine equation X4 + Y4 = 2U4 + Σni=1TiU4i, where n ≥ 1, and Ti, are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, Ti, and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can be written as sums of some biquadrates in two different ways with different coefficients.

Keywords

  • Quartic Diophantine equations
  • Biquadrates
  • Elliptic curves
  • Rank

2010 Mathematics Subject Classification

  • 11D45
  • 11D72
  • 11D25
  • 11G05
  • 14H52

References

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  3. Charmichael, R. D. (1915) Diophantine Analysis, John Wiley and Sons, Inc.
  4. Izadi, F., & Nabardi, K. (2016) Diophantine equation X4 + Y4 = 2(U4 + V4), Math. Slovaca, 66(3), 557–560.
  5. Janfada, A. S., & Shabani-solt, H. (2015) On Diophantine equation x4 + y4 = 2z4 + 2kw4, Far East Journal of Mathematical Sciences, 100(11), 1891–1899.
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  7. Sage software, Available online: http://sagemath.org.
  8. Washington, L. C. (2008) Elliptic Curves: Number Theory and Cryptography, Chapman-Hall.

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Cite this paper

APA

Abdolmalki, H. R., & Izadi, F. (2018). On a class of quartic Diophantine equations of at least five variables. Notes on Number Theory and Discrete Mathematics, 24(3), 1-9, doi: 10.7546/nntdm.2018.24.3.1-9.

Chicago

Abdolmalki, Hamid Reza, and Farzali Izadi. “On a Class of Quartic Diophantine Equations of at Least Five Variables.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 1-9, doi: 10.7546/nntdm.2018.24.3.1-9.

MLA

Abdolmalki, Hamid Reza, and Farzali Izadi. “On a Class of Quartic Diophantine Equations of at Least Five Variables.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 1-9. Print, doi: 10.7546/nntdm.2018.24.3.1-9.

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