On a class of quartic Diophantine equations

F. Izadi, M. Baghalaghdam and S. Kosari
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 1—6
DOI: 10.7546/nntdm.2021.27.1.1-6
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Authors and affiliations

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

M. Baghalaghdam
Department of Mathematics, Faculty of Science,
Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran

S. Kosari
Institute of Computing Science and Technology, Guangzhou University
Guangzhou, 510006, China


In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) { \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }, where a_i and n\geq3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a_i and n=3,4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n\geq 3, this paper will be restricted to the examples where n=3,4. Finally, we explain how to solve more general cases (n\geq 4) without giving concrete examples to case n\geq 5.


  • Quartic Diophantine equations
  • Biquadrates
  • Elliptic curves

2010 Mathematics Subject Classification

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


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Cite this paper

Izadi, F., Baghalaghdam, M., & Kosari, S. (2021). On a class of quartic Diophantine equations. Notes on Number Theory and Discrete Mathematics, 27(1), 1-6, doi: 10.7546/nntdm.2021.27.1.1-6.

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