# Pythagorean Triples

Before showing how to generate Pythagorean Triples, let us lay down a definition

A Triple comes right from the Pythagorean Theorem which states that for all integers a, b, and c, c2= a2 + b2

Definition: Integers a, b, and c, such that a2 + b2= c2

The numbers a, b, and c, are then put inside parenthesis: (a, b, c)

Notice that c is listed last and that is very important!

Example #1

32 + 42 = 52

The triple is (3, 4, 5)

Notice that 32 + 42 = 9 + 16 = 25 and 52 = 25

How would you generate another triple?

Just multiply both sides of the equation below by 22

32 + 42 = 52

22 × 32 + 22× 42 = 22× 52

( 2 × 3)2 + ( 2 × 4)2 = ( 2 × 5)2

62 + 82 = 102

You could have found the answer a lot faster than that by multiplying each number of the triple (3, 4, 5) by 2.

In general, once you have a triple, you can multiply this triple by any positive integer to generate another one

Example #2

52 + 122 = 132

The triple is (5, 12, 13)

Notice again that if 52 + 122 = 132, then 25 + 144 is indeed equal to 169

How would you generate another triple?

Just multiply both sides of the equation below by, say, 32 this time

52 + 122 = 132

32 × 52 + 32× 122 = 32× 132

( 3 × 5)2 + ( 3 × 12)2 = ( 3 × 13)2

152 + 362 = 392

Again, you could have found the answer a lot faster by multiplying each number of the triple (5, 12, 13) by 3.

Here is a little exercise: Is (4, 5, 7) is triple?

Is 42 + 52 = 72 ?

42 + 52 = 16 + 25 = 41. However, 72 = 49. So, (4, 5, 7) is not a triple

Plato, a Greek Philosopher, came up with a great formula for finding Pythagorean Triples

Plato's formula for Pythagorean Triples: (2m)2 + (m2 - 1)2 = (m2 + 1)2

To get a triple, just let m be any positive integer and do the math.

Let m = 2 for instance, we get:

(2m)2 + (m2 - 1)2 = (m2 + 1)2

(2× 2)2 + (22 - 1)2 = (22 + 1)2

(4)2 + (4 - 1)2 = (4 + 1)2

(4)2 + (3)2 = (5)2

Thus, the Pythagorean triple is (3, 4, 5)

Let m = 5 for instance, we get:

(2m)2 + (m2 - 1)2 = (m2 + 1)2

(2× 5)2 + (52 - 1)2 = (52 + 1)2

(10)2 + (25 - 1)2 = (25 + 1)2

(10)2 + (24)2 = (26)2

Thus, the triple is (10, 24, 26)

Indeed (10)2 + (24)2 = 100 + 576 = 676 and 262 = 26 × 26 = 676

## Recent Articles

1. ### Conjugate in Algebra

Oct 16, 17 09:34 AM

What is the conjugate in algebra? A concise and clear explanation.

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

## Recent Lessons

1. ### Conjugate in Algebra

Oct 16, 17 09:34 AM

What is the conjugate in algebra? A concise and clear explanation.

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests.

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.