What Are Pythagorean Triples?
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c². The most well-known example is (3, 4, 5), since 9 + 16 = 25. These triples represent the integer-sided right triangles.
Euclid's Formula
The Formula
For integers m > n > 0:
a = m² - n²
b = 2mn
c = m² + n²
b = 2mn
c = m² + n²
Primitive Triples
A triple is primitive if gcd(a, b, c) = 1. This occurs when m and n are coprime and not both odd.
gcd(m, n) = 1 and m - n is odd
Non-Primitive Triples
Multiply any primitive triple by k to get non-primitive triples.
(ka, kb, kc) for k = 2, 3, ...
Properties of Pythagorean Triples
- In every Pythagorean triple, at least one of a or b is even.
- In every primitive triple, exactly one of a or b is even.
- The hypotenuse c is always odd in a primitive triple.
- There are infinitely many Pythagorean triples.
- Every primitive triple can be generated by Euclid's formula with appropriate m and n.
Famous Pythagorean Triples
- (3, 4, 5) -- The smallest and most famous triple.
- (5, 12, 13) -- Generated by m=3, n=2.
- (8, 15, 17) -- Generated by m=4, n=1.
- (20, 21, 29) -- Nearly isosceles right triangle.