Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CUBIC FORMULA AND CUBIC CURVES

  • Woo, Sung Sik (Department of Mathematics College of Natural Science Ewha Womans University)
  • Received : 2012.03.10
  • Published : 2013.04.30
Download PDF
⟨ Previous Next ⟩

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

References

  1. N. C. Ankeny and C. A. Rogers, A conjecture of Chowla, Ann. of Math. (2) 53 (1951), 541-550. https://doi.org/10.2307/1969571
  2. H. Flanders, Generalization of a theorem of Ankeny and Rogers, Ann. of Math., (2) 57 (1953), 392-400. https://doi.org/10.2307/1969866
  3. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, New York Berlin, 1990.
  4. L. J. Mordell, Diophantine Equation, Academic Press, 1969.
  5. J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer-Verlag, New York Berlin, 2009.
  6. N. Sloane, Sequences; A013658, in "The On-Line Encyclopedia of Integer Sequences".
  7. S. S. Woo, Irreducibility of polynomials and Diophantine equations, J. Korean Math. Soc. 47 (2010), no. 1, 101-112. https://doi.org/10.4134/JKMS.2010.47.1.101
  8. B. L. Van der Waerden, Algebra, 4th ed., Frederick Ungar Publishing Co., New York, 1967.