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





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