Communications of the
Korean Mathematical Society CKMS

The problem of finding rational or integral points of anelliptic curve basically boils down to solving a cubic equation. We look closely at the cubic formula of Cardano to find a criterion for a cubic polynomial to have a rational or integral roots. Also we show that existence of a rational root of a cubic polynomial implies existence of a solution for certain Diophantine equation. As an application we find some integral solutions of some special type for $y^2=x^3+b$.

Keywords: cubic equation, rational solution, integral point of an elliptic curve