Matrix generation of Pythagorean $n$-tuples

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).