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
Full paper (PDF, 463 Kb)

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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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