Pythagorean Triples

Three whole numbers $a,b,c$ which satisfy the equation of the Pythagorean Theorem ( ${a}^{2}+{b}^{2}={c}^{2}$ ) are called Pythagorean triples . A few of the smallest ones are shown in the table below. Each Pythagorean Triple corresponds with a right triangle whose side lengths are in whole-number ratios.

 Pythagorean Triples $3,4,5$ $\begin{array}{l}{3}^{2}+{4}^{2}={5}^{2}\\ 9+16=25\end{array}$ $6,8,10$ $\begin{array}{l}{6}^{2}+{8}^{2}={10}^{2}\\ 36+64=100\end{array}$ $5,12,13$ $\begin{array}{l}{5}^{2}+{12}^{2}={13}^{2}\\ 25+144=169\end{array}$ $8,15,17$ $\begin{array}{l}{8}^{2}+{15}^{2}={17}^{2}\\ 64+225=289\end{array}$

Note that once you have one Pythagorean triple, you can get many more: just multiply all $3$ numbers by a constant. For instance, from $3,4,5$ , we get the family of triples:

$\begin{array}{l}3,4,5\\ 6,8,10\\ 9,12,15\\ 12,16,20\\ 15,20,25\end{array}$

etc.