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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Matrix generation of Pythagorean $ n$-tuples


Authors: Daniel Cass and Pasquale J. Arpaia
Journal: Proc. Amer. Math. Soc. 109 (1990), 1-7
MSC: Primary 11D09
MathSciNet review: 1000148
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Abstract: We construct, for each $ n(4 \leq n \leq 9)$, a matrix $ {A_n}$ which generates all the primitive Pythagorean $ n$-tuples $ ({x_1}, \ldots ,{x_n})$ with $ {x_n} > 1$

$\displaystyle (1)\quad x_1^2 + \cdots + x_{n - 1}^2 = x_n^2,\quad \gcd ({x_1}, \ldots ,{x_n}) = 1$

from the single $ n$-tuple $ (1,0, \ldots ,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 $ {A_n}$ to obtain another $ n$-tuple. This extends a result of Barning which presents an appropriate matrix $ {A_3}$ for the Pythagorean triples. One cannot so generate the Pythagorean $ n$-tuples if $ n \geq 10$; in fact we show the Pythagorean $ n$-tuples fall into at least $ [(n + 6)/8]$ distinct orbits under the automorphism group of (1).

References [Enhancements On Off] (What's this?)

  • [B] F. J. M. Barning, On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, Math. Centrum Amsterdam Afd. Zuivere Wisk. 1963 (1963), no. ZW-011, 37 (Dutch). MR 0190077 (32 #7491)
  • [W] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms. II, J. Reine Angew. Math. 213 (1963/1964), 122–136. MR 0155798 (27 #5732)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1000148-0
PII: S 0002-9939(1990)1000148-0
Article copyright: © Copyright 1990 American Mathematical Society