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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## Details

### 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
*

### Abstract

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) , where and 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 and , 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 , this paper will be restricted to the examples where . Finally, we explain how to solve more general cases without giving concrete examples to case .

### Keywords

- Quartic Diophantine equations
- Biquadrates
- Elliptic curves

### 2010 Mathematics Subject Classification

- 11D45
- 11D72
- 11D25
- 11G05
- 14H52

### References

- Bremner, A., Choudhry, A., & Ulas, M. (2014). Constructions of diagonal quartic and sextic surfaces with infinitely many rational points, International Journal of Number Theory, 10(7), 1675–1698.
- Baghalaghdam, M., & Izadi, F. Is the quartic Diophantine equation
*A*^{4}+*hB*^{4}=*C*^{4}+*hD*^{4}, solvable for any integer*h*?, submitted. - Dickson, L. E. (1934). History of the Theory of Numbers, Vol. II: Diophantine Analysis, G. E. Stechert Co., New York.
- Elkies, N. (1988). On
*A*^{4}+*B*^{4}+*C*^{4}=*D*^{4}. Mathematics of Computation, 51(184), 825–835. - Lander, L. J., & Parkin, T. R. (1966). Counterexamples to Euler’s conjecture on sums of like powers, Bull. Amer. Math. Soc, 72, 1079.
- Sage software, available from http://sagemath.org.

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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.