Studying of the solvability of the equations X^2-ΔY^2=-3 وَaX^2+2bXY-3aY^2=±1

Abstract

The aim of this article is studying the equations and, we proved that if equation  is unsolvable then the equation  is unsolvable, also we proved necessary and sufficient condition such that the equation is solvable when the equation  is solvable, and we proved necessary and sufficient condition such that the equation is solvable when the equation  is solvable. We proved that if  is prime number then the equation is solvable iff the equation  is solvable.