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
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Authors and affiliations

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

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


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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