The Cardano family of equations

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 5, 1999, Number 4, Pages 151–162
Full paper (PDF, 391 Kb)

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

A. G. Shannon
University of Technology, Sydney, 2007, Australia

Abstract

The polynomial expansion of the Diophantine equation x^n = (x - p)^n + (x - q)^n, p, q \in Z_{+}, n > 2, yields roots of the form ((p + q) + y) where y is a non-integer zero of a Cardano cubic polynomial of the form y^3 - 6pqy - 3pq(p + q). This is a corollary to Fermat’s Last Theorem. The characteristics of this family are illustrated for n = 3, 4, ..., 9. For n odd, y_0 can be represented by (n - 1)(2pq + e)^{\frac{1}{2}}, and for n even there are two real values of y_0, (n - 1)(2pq + e)^{\frac{1}{2}} and - (2pq + d)^{\frac{1}{2}}, where d, e are real non-integer parameters. For a given n, e is simply related to p / q, p < q, and to a parameter E which is linear in n. The corresponding curves indicate the non-integral nature of y, n > 2.

AMS Classification

  • 11C08
  • 11D41

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Cite this paper

Leyendekkers, J. V. & Shannon, A. G. (1999). The Cardano family of equations. Notes on Number Theory and Discrete Mathematics, 5(4), 151-162.

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