Algebraic and geometric analysis of a Fermat/Cardano cubic

J. V. Leyendekkers, A. G. Shannon and C. K. Wong
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 8, 2002, Number 3, Pages 85—94
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Authors and affiliations

J. V. Leyendekkers
The University of Sydney
NSW 2006, Australia

A. G. Shannon
Warrane College, The University of New South Wales, 1465, &
KvB Institute of Technology, North Sydney, 2060, Australia

C. K. Wong
Warrane College, The University of New South Wales,
Kensington, 1465

Abstract

It is shown that the functions R = - x^3 + 3(p + q)x^2 - 3(p^2 + q^2)x + (p^3 - q^3), p,q \in \mathbb{Z}_+, and R = 3x^2 - 3(2(p + q ) - 1 ) x + (3(p^2 + q^2) - 3(p + q) + 1 ) intersect at a point that is always non-integer, A geometric analysis shows that the cubic crosses the x -axis at a point, x0, that is always non-integer, with x_0 = N^\frac{1}{n} (p + q + (2pq)^\frac{1}{2}), N,n \in \mathbb{Z}_+, where N^\frac{1}{n} is obtained from the geometry of the curve. These results show that a general parameter associated with the real roots of Fermat/Cardano polynomials is a function of p, q and the geometry of the curve, which in turn yield the link with the geometry of the complex plane.

AMS Classification

  • 11C08
  • 11D41
  • 11B37

References

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Cite this paper

Leyendekkers, J., Shannon, A. & Wong C. (2002). Algebraic and geometric analysis of a Fermat/Cardano cubic. Notes on Number Theory and Discrete Mathematics, 8(3), 85-94.

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