Pythagorean Triple

A set of three positive integers (a, b, c) that satisfy the equation a^{2} + b^{2} = c^{2}, where 'c' is the greatest number, is called the **Pythagorean triple.** The smallest and best–known Pythagorean triple is (a, b, c) = (3, 4, 5), in which the sides of a right-angled triangle are in the ratio 3 : 4 : 5.

A **primitive Pythagorean triple** is a Pythagorean triple with no common factors except 1, ie., (Pythagorean triple with co-primes).

Ex: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), etc.

The Pythagorean triple (3, 4, 5) is the only primitive Pythagorean triple involving consecutive positive integers.

If the measures of the sides of any right triangle are positive integers, then the measures form a Pythagorean triple.

Ex: 5, 12, 13; 11, 60, 61; 39, 80, 89; 28, 195, 197; 68, 285, 293, etc.

**Note: **If (a, b, c) is a Pythagorean triple, then (na, nb, nc) is also a Pythagorean triple for any positive integer 'n'.