Cubic formula and cubic curves
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The problem of finding rational or integral points of an ellip-tic curve basically boils down to solving a cubic equation. We look closelyat 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 ofa rational root of a cubic polynomial implies existence of a solution forcertain Diophantine equation. As an application we find some integralsolutions of some special type for y2 = x3 + b.
Abstract
The problem of finding rational or integral points of an ellip-tic curve basically boils down to solving a cubic equation. We look closelyat 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 ofa rational root of a cubic polynomial implies existence of a solution forcertain Diophantine equation. As an application we find some integralsolutions of some special type for y2 = x3 + b.
참고문헌 (8)
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