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