[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/CKMS.2013.28.2.209

CUBIC FORMULA AND CUBIC CURVES

Woo, Sung Sik (Department of Mathematics College of Natural Science Ewha Womans University)

Publication Information

Communications of the Korean Mathematical Society / v.28, no.2, 2013 , pp. 209-224 More about this Journal

Abstract

The problem of finding rational or integral points of an elliptic 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;

Citations & Related Records

Times Cited By KSCI : 1 (Citation Analysis)

Reference
Cited By KSCI

1	S. S. Woo, Irreducibility of polynomials and Diophantine equations, J. Korean Math. Soc. 47 (2010), no. 1, 101-112. 과학기술학회마을 DOI ScienceOn
2	B. L. Van der Waerden, Algebra, 4th ed., Frederick Ungar Publishing Co., New York, 1967.
3	N. C. Ankeny and C. A. Rogers, A conjecture of Chowla, Ann. of Math. (2) 53 (1951), 541-550. DOI
4	H. Flanders, Generalization of a theorem of Ankeny and Rogers, Ann. of Math., (2) 57 (1953), 392-400. DOI
5	K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, New York Berlin, 1990.
6	L. J. Mordell, Diophantine Equation, Academic Press, 1969.
7	J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer-Verlag, New York Berlin, 2009.
8	N. Sloane, Sequences; A013658, in "The On-Line Encyclopedia of Integer Sequences".