Matrix generation of Pythagorean 𝑛-tuples

Source abstract

We construct, for each n ( 4 ≤ n ≤ 9 ) n(4 \leq n \leq 9) , a matrix A n {A_n} which generates all the primitive Pythagorean n n -tuples ( x 1 , … , x n ) ({x_1}, \ldots ,{x_n}) with x n > 1 {x_n} > 1 $$( 1 ) x 1 2 + ⋯ + x n − 1 2 = x n 2 , gcd ( x 1 , … , x n ) = 1 (1)\quad x_1^2 + \cdots + x_{n - 1}^2 = x_n^2,\quad \gcd ({x_1}, \ldots ,{x_n}) = 1$$ from the single n n -tuple ( 1 , 0 , … , 0 , 1 ) (1,0, \ldots ,0,1) . Once a particular n n -tuple is generated, one permutes the first n − 1 n - 1 coordinates and/or changes some of their signs, and applies A n {A_n} to obtain another n n -tuple. This extends a result of Barning which presents an appropriate matrix A 3 {A_3} for the Pythagorean triples. One cannot so generate the Pythagorean n n -tuples if n ≥ 10 n \geq 10 ; in fact we show the Pythagorean n n -tuples fall into at least [ ( n + 6 ) / 8 ] [(n + 6)/8] distinct orbits under the automorphism group of (1).