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
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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.
- Quartic Diophantine equations
- Elliptic curves
2010 Mathematics Subject Classification
- 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 X4 + Y4 = 2(U4 + V4), Math. Slovaca, 66(3), 557–560.
- Janfada, A. S., & Shabani-solt, H. (2015) On Diophantine equation x4 + y4 = 2z4 + 2kw4, 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.
Cite this paper
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.