Animation demonstrating the simplest case of the Pythagorean Triple: 32 + 42 = 52.
There are 52 Pythagorean triples with c ≤ 100 out of which 16 are primitive triples. Click for details.
A set of three positive integers (a, b, c) that satisfy the equation a2 + b2 = c2, 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, i.e., (Pythagorean triple with co-primes). Ex: (3, 4, 5), (5, 12, 13), (8, 15, 17), etc. (6, 8, 10) is not a primitive triple. 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-angled triangle are positive integers, then the measures form a Pythagorean triple. Ex: 11, 60, 61; 39, 80, 89; 28, 195, 197; etc. If (a, b, c) is a Pythagorean triple, then (na, nb, nc) is also a Pythagorean triple for any positive integer 'n'.
Euclid's formula for generating Pythagorean triples: This is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers 'm' and 'n' with m > n. The formula states that: "for any two positive integers 'm' and 'n' with m > n, a = m2 – n2, b = 2mn, c = m2 + n2 form a Pythagorean triple."
Note: The Pythagorean triple generated by Euclid's formula is primitive if and only if 'm' and 'n' are co-primes and (m – n) is odd. If both 'm' and 'n' are odd, then 'a', 'b', and 'c' will be even, and so the triple will not be primitive; however, dividing 'a', 'b', and 'c' by 2 will yield a primitive triple if 'm' and 'n' are co-prime.