One of the coolest old bits of math is the Pythagorean theorem. For any right triangle, the sum of the squares of the length of the legs equals the square of the length of the hypotenuse. That is a2 + b2 = c2.
The formula wasn’t invented by Pythagorus, but was known as early as 1800 BCE, in Babylon. What Pythagorus adds is the focus on whole number solutions, like 3, 4, 5. These are called Pythagorean triples, something important to Pythagorus because he believed that the essence of the world was whole numbers. It’s an approach that also is found with Democritus and Epicurus. Democracy, for example being, the idea that the whole of the national sprit is based on the will of unit individuals rather than the will of a single mystic who alone sees the political soul.
There is a formula for finding Pythagorean triples that was discovered, or first presented, by Euclid. Pick any two positive whole numbers, m and n where m>n. For any value of m and n he showed that there is a pythagorean triple where the hypotenuse equals m2+n2 and where one side equals m2-n2 and the other side is 2mn. For example, if you pick m=3 and n=2, you’ll find there is a triple with hypotenuse 13, and with legs 5 and 12. It’s a very cool formula, I think. Like pythagorus, I find a certain magic in Pythagorean triples.
The proof of this method is to note that (m2-n2)2 = m4 + n4 – 2m22n2. Now note that this is very similar to the formula for (m2+n2)2 = m4 + n4 + 2m2n2. We see that (m2+n2)2 = (m2-n2)2 + 4m2n2. These are all whole numbers since m and n are whole numbers, and they have the form for a pythagorus triple. This formula gives you all the “fundamental “primitive” triples, plus some others. Once you have these, you can find the remaining ripples by multiplying these by 2, 3, 4, etc.
There is another nice formula for whole-number triangles in 3 space, useful for boat building and the like — if you go to the shop and ask for someone to cut wood, they generally want to cut to whole numbers, usually whole inches, 5″ by 12″. With pythagorean triples, there is a better chance that your pieces will fit together. (I was designing and building boats a few posts back) Maybe I’ll go into it in another post.
There are 16 primitive Pythagorean triples of numbers up to 100.
| (3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
| (20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
| (11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
| (13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
Robert Buxbaum, June 1, 2026