# Pythagorean Triples

## What is a Pythagorean triple?

Some numbers seem to work perfectly in the Pythagorean Theorem, like **3**, **4**, and **5**, which is ${3}^{2}+{4}^{2}={5}^{2}$. Sets of positive, whole numbers that work in the Pythagorean Theorem are calledÂ **Pythagorean triples**.

For three positive integers to be Pythagorean triples, they must work in the Pythagorean Theorem's formula:

In the Pythagorean Theorem's formula,Â * a*Â andÂ

*Â are legs of a right triangle, andÂ*

**b***Â is the hypotenuse.*

**c**Only positive integers can be Pythagorean triples. The smallest Pythagorean triple is our example: (**3**, **4**, and** 5**).

A quick way to find more Pythagorean triples is to multiply all the original terms by another positive integer:

## Pythagorean triples

Pythagorean triples are relatively prime.Â **Relatively prime**Â means they have no common divisor other than **1**, even if the numbers are not prime numbers, like **14** and **15**. The number 14 has factors **1**, **2**, **7**, and **14**; the number **15** has factors **1**,** 3**,** 5**, and **15**. Their only common factor is **1**.

### Primitive Pythagorean triples

A set of numbers is considered to be aÂ **primitive Pythagorean triple**Â if the three numbers have no common divisor other than** 1**.

Our first Pythagorean triple is primitive, since (**3**, **4**, and** 5**) have no common divisors other than **1**. Our fifth set from our example above, however, is not primitive (it is imprimitive) because each value forÂ ** a**,Â

**, andÂ**

*b***Â of the right triangle is a multiple of**

*c***5**.

### How to find Pythagorean triples

Here's how to find Pythagorean triples in three easy steps:

Pick an even number to be the longer leg's length.

Find a prime number one greater than that even number, to be the hypotenuse.

Calculate the third value to find the Pythagorean triple.

## Pythagorean triples formula

Suppose you pick **12** as the length of a leg, knowing **13** is an adjacent prime number. Use these two as part of the Pythagorean Theorem to complete your primitive Pythagorean triple:

Subtract the value ofÂ ${b}^{2}$Â from both sides:

So our primitive Pythagorean triple is (**5**, **12**, **13**)! You can quickly generate iterations of that (none of which will be primitive) using the same multiplying method we used before: (**10**,** 24**, **26**); (**15**, **36**, **39**); and so on.

## Pythagorean triples examples

Generate a primitive Pythagorean triple based on the prime number **41**. What even number is adjacent to and smaller than **41**?

Did you getÂ **a=9**,Â **b=40**, andÂ **c=41**? The primitive Pythagorean triple is (**9**,** 40**, **41**).

## Generating Pythagorean triples

You can come up with your own Pythagorean triples. Label two positive integersÂ * m*Â andÂ

*, ensuringÂ*

**n****m>n**. Then, for each side of a right triangle:

### Formula for generating Pythagorean triples

Generate a set of Pythagorean triples forÂ **m=6**Â andÂ **n=5**Â using our formula:

Let's solve for * a*:

Now, let's solve for * b*:

And finally, * c*:

Our Pythagorean triple is (**11**, **60**,** 61**). This method catches primitive and imprimitive Pythagorean triples.

## Pythagorean triples list

I know many of you are eager to get your hands on a Pythagorean Triples list already pre-computed for you. Here's a list of some common ones:

3, 4, 5

5, 12, 13

6, 8, 10

7, 24, 25

8, 15, 17

9, 12, 15

9, 40, 41

10, 24, 26

12, 16, 20

12, 35, 37

14, 48, 50

15, 20, 25

15, 36, 39

16, 30, 34

18, 24, 30

20, 21, 29

21, 28, 35

24, 32, 40

27, 36, 45

30, 40, 50

## Lesson summary

By watching the video, reading the lesson, and studying the formulas, you have learned how to identify a Pythagorean triple, apply and use the Pythagorean Theorem (**32Â + 42Â = 52**), classify a Pythagorean triple as either primitive or imprimitive, and use both the Pythagorean Theorem and another method to find Pythagorean triples.

### What you learned:

After reading the lesson and studying the drawings you will be able to:

Identify a Pythagorean triple

Apply the Pythagorean Theorem

Categorize a Pythagorean triple as either primitive or

**imprimitive**(not primitive)Use the Pythagorean Theorem and another method to find Pythagorean triples