Pythagorean Triples

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

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

A Pythagorean triple is a set of three whole numbers a,b, and c bigger than zero 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!

Pythagorean Triples

How find Pythagorean triples

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 and the Pythagorean triple is (6,8,10)

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 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's formula for Pythagorean Triples:

Plato, a Greek Philosopher, came up with a great formula for finding 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. Find the Number of Combinations

    Jul 30, 21 06:15 AM

    Learn quickly how to find the number of combinations with this easy to follow lesson.

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.