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Matrix Generation of Pythagorean n-Tuples

Daniel Cass and Pasquale J. Arpaia
Proceedings of the American Mathematical Society
Vol. 109, No. 1 (May, 1990), pp. 1-7
DOI: 10.2307/2048355
Stable URL: http://www.jstor.org/stable/2048355
Page Count: 7
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Matrix Generation of Pythagorean n-Tuples
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Abstract

We construct, for each n(4 ≤ n ≤ 9), a matrix An which generates all the primitive Pythagorean n-tuples (x1,..., xn) with $x_n > 1$ \begin{equation*}\tag{1} x^2_1 + \cdots + x^2_{n - 1} = x^2_n,\quad \operatorname{gcd}(x_1,\ldots, x_n) = 1 \end{equation*} from the single n-tuple (1, 0,..., 0, 1). Once a particular n-tuple is generated, one permutes the first n - 1 coordinates and/or changes some of their signs, and applies An to obtain another n-tuple. This extends a result of Barning which presents an appropriate matrix A3 for the Pythagorean triples. One cannot so generate the Pythagorean n-tuples if n ≥ 10; in fact we show the Pythagorean n-tuples fall into at least [(n + 6)/8 ] distinct orbits under the automorphism group of (1).

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