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
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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 .
- Quartic Diophantine equations
- Elliptic curves
2010 Mathematics Subject Classification
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- Sage software, available from http://sagemath.org.
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.