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, c

Definition: Integers a, b, and c, such that a

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!

3

The triple is (3, 4, 5)

Notice that 3

How would you generate another triple?

Just multiply both sides of the equation below by 2

3

2

( 2 × 3)

6

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

5

The triple is (5, 12, 13)

Notice again that if 5

How would you generate another triple?

Just multiply both sides of the equation below by, say, 3

5

3

( 3 × 5)

15

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 4

4

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

Plato's formula for Pythagorean Triples: (2m)

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

Let m = 2 for instance, we get:

(2m)

(2× 2)

(4)

(4)

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

Let m = 5 for instance, we get:

(2m)

(2× 5)

(10)

(10)

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

Indeed (10)