Abstract
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.
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The author is grateful to an anonymous referee for his/her appropriate comments.
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Dedicated to the 60th birthday of Sibylle von Burg-Baldini on July 29, 2016, whose ZH car number is 1729
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Hürlimann, W. Permutation invariant properties of primitive cubic quadruples. Ramanujan J 43, 649–662 (2017). https://doi.org/10.1007/s11139-016-9798-9
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DOI: https://doi.org/10.1007/s11139-016-9798-9
Keywords
- Cubic quadruple
- Sum of two cubes
- Taxicab number
- Cabtaxi number
- Euler rational parameterization
- Quadratic Hopf map
- Integer triple ordering