Matrix generation of Pythagorean $n$-tuples
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- by Daniel Cass and Pasquale J. Arpaia PDF
- Proc. Amer. Math. Soc. 109 (1990), 1-7 Request permission
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 ${x_n} > 1$ \[ (1)\quad x_1^2 + \cdots + x_{n - 1}^2 = x_n^2,\quad \gcd ({x_1}, \ldots ,{x_n}) = 1\] 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
- 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 190077
- C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms. II, J. Reine Angew. Math. 213 (1963/64), 122–136. MR 155798, DOI 10.1515/crll.1964.213.122
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 1-7
- MSC: Primary 11D09
- DOI: https://doi.org/10.1090/S0002-9939-1990-1000148-0
- MathSciNet review: 1000148