Cubic formula and cubic curves

Abstract

The problem of finding rational or integral points of an ellip-tic 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 polyno-mial 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 y2 = x3 + b. © 2013 The Korean Mathematical Society.