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Parametrization of Pythagorean triples by a single triple of polynomials. (English) Zbl 1215.11025
Summary: It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.
MSC:
11D09Quadratic and bilinear diophantine equations
11D85Representation problems of integers
11C08Polynomials (number theory)
13F20Polynomial rings and ideals
References:
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[2], Lecture notes in pure and applied mathematics 171 (1995)
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[4]Cahen, P. -J.; Chabert, J. -L.: Integer-valued polynomials, Mathematical surveys and monographs 48 (1997) · Zbl 0884.13010
[5]Chapman, S.; Mcclain, B. A.: Irreducible polynomials and full elasticity in rings of integer-valued polynomials, J. algebra 293, 595-610 (2005) · Zbl 1082.13001 · doi:10.1016/j.jalgebra.2005.01.026
[6]S. Frisch, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra. (in press) · Zbl 1209.11038 · doi:10.1080/00927870701776938
[7]Hlawka, E.; Schoißengeier, J.: Zahlentheorie. eine einführung, (1979)
[8]L. Vaserstein, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Ann. Math. (in press) · Zbl 1221.11082 · doi:10.4007/annals.2010.171.979 · doi:http://annals.princeton.edu/annals/2010/171-2/p07.xhtml