Pythagorean triples: Baudhayana triples

Right triangles with whole-number sides

Most right triangles have at least one side that is not a whole number: sides 1 and 1 give a hypotenuse of √2. But some have three whole-number sides. In his Shulbasutra (verse 1.13), Baudhayana listed six: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (15, 36, 39) and (12, 35, 37). Three counting numbers with a² + b² = c² are called Baudhayana triples, and also Baudhayana-Pythagoras triples or Pythagorean triples. Check one: 5² + 12² = 25 + 144 = 169 = 13².

Scaling a triple

Is (30, 40, 50) a triple? Yes: 900 + 1600 = 2500. So is (300, 400, 500). In general (3k, 4k, 5k) works for every counting number k, since 9k² + 16k² = 25k². The same argument works for any triple: if a² + b² = c², then (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)². A triple multiplied like this is a scaled version, so there are infinitely many triples. The triples using numbers up to 20 are (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17) and (12, 16, 20).

Primitive triples

A triple whose numbers have no common factor greater than 1 is primitive. (3, 4, 5), (5, 12, 13) and (8, 15, 17) are primitive; (9, 12, 15) is not, because 3 divides all three numbers. Dividing a non-primitive triple by its common factor f gives another triple, since a² + b² = c² still holds after dividing everything by f²: (9, 12, 15) becomes (3, 4, 5). So every triple is either primitive or a scaled version of a primitive one, and finding all primitive triples means finding them all.

Making triples from odd squares

Add odd numbers in turn: 1 + 3 = 4 = 2², 1 + 3 + 5 = 9 = 3², 1 + 3 + 5 + 7 = 16 = 4². The first n odd numbers always add up to n², and the odd number in position n is 2n − 1. So the first n − 1 odd numbers make (n − 1)², and (n − 1)² + (2n − 1) = n². Now pick n so that 2n − 1 is itself a square. For 9 = 3², the 5th odd number, n = 5 and 4² + 3² = 5². For 25 = 5², the 13th, n = 13 and 12² + 5² = 13². The odd squares 49, 81 and 121 give (7, 24, 25), (9, 40, 41) and (11, 60, 61).

These triples are all primitive, because the hypotenuse n and the side n − 1 are consecutive numbers, which share no factor. But the method misses some primitive triples, such as (8, 15, 17) and (20, 21, 29), where no side is one less than the hypotenuse.

What about cubes?

Squares can be sums of two squares in infinitely many ways. In the 17th century the French mathematician Fermat claimed that no cube is a sum of two cubes, nor any higher power the sum of two such powers: xⁿ + yⁿ = zⁿ has no solution in counting numbers when n is more than 2. His proof was never found. After more than 300 years of attempts, Andrew Wiles, who had read about the problem at the age of 10, proved this Fermat’s Last Theorem in 1994. See the theorem, the square root of 2, the Chapter 2 notes and the NCERT solutions.