# Pythagorean triple

A **Pythagorean triple** is a triple of positive integers, such that . Pythagorean triples arise in geometry as the side-lengths of right triangles.

## Contents

## Common Pythagorean Triples

These are some common Pythagorean triples: *=Primitive (see below) (3, 4, 5)*

(5, 12, 13)*

(6, 8, 10)

(7, 24, 25)*

(8, 15, 17) *

(9, 12, 15)

(9, 40, 41) *

(20, 21, 29)*

(11, 60, 61)*

(13, 84, 85)*

(12, 35, 37) *

(16, 63, 65) *

(36, 77, 85)*

(33, 56, 65) *

(39, 80, 89)*

(28, 45, 53)*

(48, 55, 73) *

(65, 72, 97)*

## Pythagorean Triangles

Each positive integer solution of the diophantine equation defining the Pythagorean triples satisfies . Thus, any triple of positive integers satisfying this equation also satisfies the triangle inequality, so the solutions correspond to right triangles with integral side lengths.

## Primitive Pythagorean Triples

A Pythagorean triple is called *primitive* if its three members have no common divisors, so that they are relatively prime. All triples listed above are primitive. Integral multiples of Pythagorean triples will also satisfy , but they will not form primitive triples. For example, all triples of integers of the form , such as , are Pythagorean triples.

### General Form of Primitive Pythagorean Triples

**Theorem.** A triple of integers is a primitive Pythagorean triple if and only if it may be written in the form or , where are relatively prime positive integers of different parity.

#### Proof

Let be a primitive Pythagorean triple. If and both odd, then we must have which is a contradiction, since 2 is not a square mod 4. Hence at least one of and , say , is even. Then must be odd, since and must be relatively prime. It follows that is odd as well. It follows that the numbers and are positive integers. These positive integers must be relatively prime, since any common divisor of and must divide both and . Since and , it follows that Since must be an integer and and are relatively prime, it follows that and are perfect squares. Hence we may denote and for integers and . Since is odd, it follows that and must have different parity, so and have different parity. Finally, we observe that so any triple of the form specified in the theorem is a Pythagorean triple; it must furthermore be a primitive Pythagorean triple, since any common factor of and (both of which are odd integers, since and have different parity) must also be a factor of both and , which are integers with no common factor greater than 2.