Pythagorean triple preserving matrices

From Table (1), let

$$\mathbf{C}= \begin{pmatrix} 3&4&5\\ 21&20&29\\ 119&120&169 \end{pmatrix}$$   and$$\quad \mathbf{D}= \begin{pmatrix} 21&20&29\\ 119&120&169\\ 697&696&985 \end{pmatrix}$$

So, solving for $\mathbf{A}$ in the matrix equation $\mathbf{C}\mathbf{A}=\mathbf{D}$ , we find that

$$\mathbf{A}= \begin{pmatrix} 1&2&2\\ 2&1&2\\ 2&2&3 \end{pmatrix}.$$

Let $a^2+b^2=c^2$ . Then $\begin{pmatrix}a&b&c\end{pmatrix}\mathbf{A}=\begin{pmatrix}u&v&w\end{pmatrix}$ where $u^2+v^2=w^2$ , and $u-v=b-a$ . Furthermore,

$$\begin{pmatrix} m^2-n^2&2mn&m^2+n^2 \end{pmatrix}\begin{pmatrix} ... ...&3 \end{pmatrix}= \begin{pmatrix} \mu^2-\nu^2&2\mu\nu&\mu^2+\nu^2 \end{pmatrix}$$

where $\mu=2m+n$ and $\nu=m$ . Therefore, if $(a,b,c)$ is a primitive Pythagorean triple then so is $(u,v,w)$ . Thus, if $n\in \mathbb{Z}^+$ , then $\begin{pmatrix}1&0&1\end{pmatrix}\mathbf{A}^n=\begin{pmatrix}u_n&v_n&w_n\end{pmatrix}$ gives the $n^{th}$ primitive Pythagorean triple such that $u_n-v_n=\pm 1$ .

We have, using matrix $\mathbf{A}$ , if $(a,b,c)$ is a primitive Pythagorean triple then

$$\begin{pmatrix}-a&b&c \end{pmatrix}\begin{pmatrix} 1&2&2\\ 2&1&2... ...ix} -1&-2&-2\\ 2&1&2\\ 2&2&3 \end{pmatrix}=\begin{pmatrix}r&s&t \end{pmatrix}$$

where $r, s$ and $t$ are pairwise relatively prime, $r^2+s^2=t^2$ , and

$$t-r=(-2a+2b+3c)-(-a+2b+2c)=c-a.$$

Also, $r, s$ , and $t$ are positive integers, since $c>a$ . Hence $(r,s,t)$ is a primitive Pythagorean triple.

Similarly, if $(a,b,c)$ is a primitive Pythagorean triple then

$$\begin{pmatrix}a&b&c \end{pmatrix}\begin{pmatrix} 1&2&2\\ -2&-1&-2\\ 2&2&3 \end{pmatrix}=\begin{pmatrix}x&y&z \end{pmatrix}$$

where $(x,y,z)$ is a primitive Pythagorean triple such that $z-y=c-b$ .

Let

$$\mathbf{D}=\begin{pmatrix} -1&-2&-2\\ 2&1&2\\ 2&2&3 \end{pmatrix}$$   and$$\quad \mathbf{U}=\begin{pmatrix} 1&2&2\\ -2&-1&-2\\ 2&2&3 \end{pmatrix}.$$

Joe Roberts² has shown that $(a,b,c)$ is a Primitive Pythagorean triple if and only if

$$\begin{pmatrix}a&b&c \end{pmatrix}=\begin{pmatrix}3&4&5 \end{pmatrix}\mathbf{\mathfrak{M}}$$

where $\mathbf{\mathfrak{M}}$ is a finite product of the matrices $\mathbf{U}, \mathbf{A}$ , and $\mathbf{D}$ .

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