Triples, Pathagorean Triangles
Pythagorean Triples are right triangles with all three sides of integer length. The number of such triangles is infinite. An algorithm for finding these triangles, attributed to Euclid, is the following.
Choose any 2 integers, $m$ and $n$, such that $n>m$.
$$\text{Side }a=n^{2}-m^{2}$$
$$\text{Side }b=2\cdot m\cdot n$$
$$\text{Side }c=n^{2}+m^{2}$$
Example: Use the algorithm to find a Pythagorean Triangle when $n=17$ and $m=12$.
Answer:
$$a=17^2 - 12^2 =289-144=145$$
$$b=2\cdot 17\cdot 12 = 408$$
$$c=17^2+12^2 = 433$$
Check:
$$a^2 + b^2 = c^2$$
$$145^2 + 408^2 =187489= 433^2$$
Other than for having fun playing with numbers, an appreciation for this algorithm requires an introduction to elementary number theory.