zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples. (English) Zbl 1273.11053
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.
11D09Quadratic and bilinear diophantine equations
11D72Equations in many variables
Full Text: DOI arXiv
[1] Carmichael, R. D.: Diophantine analysis, (1915) · Zbl 45.0283.11 · http://resolver.library.cornell.edu/math/2143901
[2] Conway, John H.; Smith, Derek A.: On quaternions and octonions. Their geometry, arithmetic, and symmetry, (2003) · Zbl 1098.17001
[3] Descartes, R.: Correspondance IV, Oeuvres de Descartes (1901) · Zbl 32.0006.02
[4] Dickson, Leonard Eugene: Algebras and their arithmetics, (1960) · Zbl 0086.25602
[5] Eriksson, Nicholas; Lagarias, J. C.: Apollonian circle packings: number theory. II. spherical and hyperbolic packings, Ramanujan J. 14, No. 3, 437-469 (2007) · Zbl 1165.11057 · doi:10.1007/s11139-007-9052-6
[6] Frisch, Sophie: Remarks on polynomial parametrization of sets of integer points, Comm. algebra 36, No. 3, 1110-1114 (2008) · Zbl 1209.11038 · doi:10.1080/00927870701776938
[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 · doi:10.1016/j.jpaa.2007.05.019
[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 · doi:10.1016/S0022-314X(03)00015-5
[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, No. 1, 1-36 (2006) · Zbl 1085.52011 · doi:10.1007/s00454-005-1195-x
[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, No. 1, 37-72 (2006) · Zbl 1085.52012 · doi:10.1007/s00454-005-1197-8
[11] Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.: Beyond the Descartes circle theorem, Amer. math. Monthly 109, No. 4, 338-361 (2002) · Zbl 1027.51022 · doi:10.2307/2695498
[12] Northshield, S.: On integral apollonian circle packings, J. number theory 119, No. 2, 171-193 (2006) · Zbl 1102.52301 · doi:10.1016/j.jnt.2005.10.003
[13] Soddy, F.: The kiss precise, Nature 137, No. 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, No. 5, 993-1054 (1976) · Zbl 0338.13015
[15] Vaserstein, L. N.: Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Ann. of math. 171, No. 2, 979-1009 (2010) · Zbl 1221.11082 · doi:10.4007/annals.2010.171.979 · http://annals.princeton.edu/annals/2010/171-2/p07.xhtml