Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples
Number Theory
2012-01-04 v2
Abstract
A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. For n=4 and n=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 quintuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficients.
Keywords
Cite
@article{arxiv.1106.3493,
title = {Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples},
author = {Sophie Frisch and Leonid Vaserstein},
journal= {arXiv preprint arXiv:1106.3493},
year = {2012}
}
Comments
11 pages; to appear in J. Pure Appl. Algebra; June 28 2011: corrected typo in Thm 1 (plus instead of minus in first coord)