Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples

Details Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples

2011-06-17

arxiv.org/abs/1106.3493v2

J. Pure Appl. Algebra 216 (1) (2012) 184-191

Mathematics

Number Theory

11 pages; to appear in J. Pure Appl. Algebra; June 28 2011: corrected typo in Thm 1 (plus instead of minus in first coord)

Scientific paper

10.1016/j.jpaa.2011.06.002

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.

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