The Diophantine Equation ax⁶ + by³ + cz² = 0 in Gaussian Integers

Abstract

In this article, we will examine the Diophantine equation $ax^6+by^3+cz^2=0$, for arbitrary rational integers a, b, and c in Gaussian integers and find all the solutions of this equation for many different values of a, b, and c. Moreover, two equations of the type $x^6{\pm}iy^3+z^2=0$, and $x^6+y^3{\pm}wz^2=0$ are also discussed, where i is the imaginary unit and w is a third root of unity.

Keywords

Diophantine equations;Gaussian integers;Elliptic curves;ranks;torsion group

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References

1. H. Cohen, Number Theory Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, Springer-Verlag, ISBN 978-0-387-49922-2. page 389.
2. J. Cremona, mwrank program, Available at http://maths.nottingham.ac.uk/personal/jec/ftp/progs/.
3. E. Lampakis, In Gaussian integers, $x^3+y^3=z^3$ has only trivial solutions, Elec. J. Comb. N. T, 8(2008), #A32.
4. M. A. Kenku, F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J., 109(1988), 125-149.
5. T. Nagell, Introduction to number theory, Chelsea Publ. Comp., New York, 1981.
6. F. Najman, The diophantine equation $x^4{\pm}y^4=iz^2$ in Gaussian integers, Amer. Math. Monthly, 117(2010), 637-641. https://doi.org/10.4169/000298910X496769
7. F. Najman, Torsion of elliptic curves over quadratic cyclotomic fields, Math. J. Okayama Univ..
8. F. Najman, Complete classification of torsion of elliptic curves over quadratic cyclotomic fields, J. Number Theory.
9. J. H. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathemathics, Springer-Verlag, New York, 1992.
10. U. Schneiders and H. G. Zimmer, The rank of elliptic curves upon quadratic extensions, Computational Number Theory (A. Petho, H. C. Williams, H. G. Zimmer, eds.), de Gruyter, Berlin, 1991, 239-260.