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

  • IZADI, FARZALI (Department of Mathematics, Faculty of Science, Urmia university) ;
  • KHOSHNAM, FOAD (Department of Mathematics, Faculty of Science, Urmia university)
  • Received : 2014.07.16
  • Accepted : 2015.02.21
  • Published : 2015.09.23

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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