Baudhāyana triples

Most right triangles have an awkward hypotenuse: √288, √50, and so on. But some have three whole-number sides, and Baudhāyana lists several in Verse 1.13: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (12, 35, 37), (15, 36, 39).

Whole numbers that satisfy a² + b² = c² are called Baudhāyana triples (also Baudhāyana-Pythagoras triples, Pythagorean triples, or right-angled triangle triples).

Are there only a few? List the ones with numbers up to 20: (3, 4, 5), (6, 8, 10), (9, 12, 15), (12, 16, 20). Look closely: every one is (3, 4, 5) multiplied through by a whole number.

That suggests a guess: (3k, 4k, 5k) is always a triple. Check it: (3k)² + (4k)² = 9k² + 16k² = 25k² = (5k)². True, so there are infinitely many triples.

The same works from any triple: if a² + b² = c², then (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)². We call (ka, kb, kc) a scaled version.

So the interesting triples are the ones that aren't scaled copies of something smaller. A triple with no common factor greater than 1 is primitive. (3, 4, 5) is primitive; (9, 12, 15) isn't, being 3 times it. Find all the primitives and you've found them all.

How do you find new primitives? Use a fact from Part 1: the sum of the first n odd numbers is n², and the nth odd number is 2n − 1. So (n − 1)² + (2n − 1) = n².

Now the trick. If that odd number 2n − 1 happens itself to be a square, the equation becomes square + square = square:

Every odd square gives a new triple. (These always have a hypotenuse just one more than a leg, so triples like (8, 15, 17) need other methods.)