# Permutation invariant properties of primitive cubic quadruples

01.08.2017
Published in:
• The Ramanujan Journal. - 2017, vol. 43, no. 3, p. 649–662
##### Integer triple ordering
English Based on a specific quadratic Hopf map between the Euclidean spaces of dimension four and three that is associated with Euler’s complete rational parameterization of the four cubes equation, we study the permutation invariant properties of the primitive integer cubic quadruples that solve this equation. Fixing the coordinate with maximum height and taking it positive, our main result describes the six positive primitive triples that leave it invariant under the inverted cubic map to this Hopf map and permute the remaining integer coordinates. The obtained invariant primitive triples are ordered in the so-called integer triple ordering, so that the minimum triple with respect to this ordering determines each primitive cubic quadruple uniquely. Implications for the counting and enumeration of all primitive cubic quadruples are mentioned. A list of all primitive cubic quadruples with positive maximum height below 100 and their minimum invariant triples is given. The relationship with the famous Taxicab and Cabtaxi numbers is also explained.
Faculty
Faculté des sciences et de médecine
Department
Département de Mathématiques
Language
• English
Classification
Mathematics