Frisch, Sophie; Vaserstein, Leonid Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples. (English) Zbl 1273.11053 J. Pure Appl. Algebra 216, No. 1, 184-191 (2012). Summary: For \(n=4\) or 6, the Pythagorean \(n\)-tuples admit a parametrization by a single \(n\)-tuple of polynomials with integer coefficients (which is impossible for \(n=3\)). For \(n=5\), there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean 5-tuples (similar to the case \(n=3\)). Pythagorean quadruples are closely related to (integer) Descartes quadruples, which we also parametrize by a Descartes quadruple of polynomials with integer coefficients. MSC: 11D09 Quadratic and bilinear Diophantine equations 11D72 Diophantine equations in many variables PDFBibTeX XMLCite \textit{S. Frisch} and \textit{L. Vaserstein}, J. Pure Appl. Algebra 216, No. 1, 184--191 (2012; Zbl 1273.11053) Full Text: DOI arXiv References: [1] Carmichael, R. D., Diophantine Analysis (1915), John Wiley & Sons: John Wiley & Sons New York · JFM 45.0283.11 [2] Conway, John H.; Smith, Derek A., On Quaternions and Octonions. Their Geometry, Arithmetic, and Symmetry (2003), Peters · Zbl 1098.17001 [3] Descartes, R., (Adam, C.; Tannery, P., Correspondance IV. Correspondance IV, OEuvres de Descartes (1901), Leopold Cert: Leopold Cert Paris) [4] Dickson, Leonard Eugene, Algebras and their Arithmetics (1960), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0086.25602 [5] Eriksson, Nicholas; Lagarias, J. C., Apollonian circle packings: number theory. II. Spherical and hyperbolic packings, Ramanujan J., 14, 3, 437-469 (2007) · Zbl 1165.11057 [6] Frisch, Sophie, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra, 36, 3, 1110-1114 (2008) · Zbl 1209.11038 [7] Frisch, Sophie; Vaserstein, Leonid, Parametrization of Pythagorean triples by a single triple of polynomials, J. Pure Appl. Algebra, 212, 271-274 (2008) · Zbl 1215.11025 [8] Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, C. R., Apollonian circle packings: number theory, J. Number Theory, 100, 1-45 (2003) · Zbl 1026.11058 [9] Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, Catherine H., Apollonian circle packings: geometry and group theory. II. Super-Apollonian group and integral packings, Discrete Comput. Geom., 35, 1, 1-36 (2006) · Zbl 1085.52011 [10] Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; Yan, Catherine H., Apollonian circle packings: geometry and group theory. III. Higher dimensions, Discrete Comput. Geom., 35, 1, 37-72 (2006) · Zbl 1085.52012 [11] Lagarias, J. C.; Mallows, C. L.; Wilks, A. R., Beyond the Descartes circle theorem, Amer. Math. Monthly, 109, 4, 338-361 (2002) · Zbl 1027.51022 [12] Northshield, S., On integral Apollonian circle packings, J. Number Theory, 119, 2, 171-193 (2006) · Zbl 1102.52301 [13] Soddy, F., The kiss precise, Nature, 137, 3477, p1021 (1936) [14] Vaserstein, L. N.; Suslin, A. A., Serre’s problem on projective modules over polynomial rings and algebraic \(K\)-theory, Izv. Akad. Nauk, Ser. Mat., 40, 5, 993-1054 (1976), Math. USSR Izv. 10 (5) 937-1001 · Zbl 0338.13015 [15] Vaserstein, L. N., Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Ann. of Math., 171, 2, 979-1009 (2010) · Zbl 1221.11082 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.