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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## Details

### 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 *X*^{4} + *Y*^{4} = 2*U*^{4} + Σ^{n}_{i=1}*T _{i}U*

^{4}

_{i}, where

*n*≥ 1, and

*T*, 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

_{i}*n*,

*T*, 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.

_{i}### Keywords

- Quartic Diophantine equations
- Biquadrates
- Elliptic curves
- Rank

### 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.
- Bremner, A. (1987) On Euler’s quartic surface, Math. Scand, 61, 165–180.
- Charmichael, R. D. (1915) Diophantine Analysis, John Wiley and Sons, Inc.
- Izadi, F., & Nabardi, K. (2016) Diophantine equation X
^{4}+ Y^{4}= 2(U^{4}+ V^{4}), Math. Slovaca, 66(3), 557–560. - Janfada, A. S., & Shabani-solt, H. (2015) On Diophantine equation x
^{4}+ y^{4}= 2z^{4}+ 2kw^{4}, Far East Journal of Mathematical Sciences, 100(11), 1891–1899. - Mordell, L. J. (1969) Diophantine Equations, Academic Press, Inc.
- Sage software, Available online: http://sagemath.org.
- Washington, L. C. (2008) Elliptic Curves: Number Theory and Cryptography, Chapman-Hall.

## Related papers

## Cite this paper

APAAbdolmalki, 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.

ChicagoAbdolmalki, 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.

MLAAbdolmalki, 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.