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, c
^{2}= a
^{2} + b
^{2}
Definition: Integers a, b, and c, such that a
^{2} + b
^{2}= c
^{2}
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 3
^{2} + 4
^{2} = 5
^{2}
The triple is (3, 4, 5)
Notice that 3
^{2} + 4
^{2} = 9 + 16 = 25 and 5
^{2} = 25
How would you generate another triple?
Just multiply both sides of the equation below by 2
^{2}
3
^{2} + 4
^{2} = 5
^{2}
2
^{2} × 3
^{2} + 2
^{2}× 4
^{2} = 2
^{2}× 5
^{2}
( 2 × 3)
^{2} + ( 2 × 4)
^{2} = ( 2 × 5)
^{2}
6
^{2} + 8
^{2} = 10
^{2}
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
5
^{2} + 12
^{2} = 13
^{2}
The triple is (5, 12, 13)
Notice again that if 5
^{2} + 12
^{2} = 13
^{2}, then 25 + 144 is indeed equal to 169
How would you generate another triple?
Just multiply both sides of the equation below by, say, 3
^{2} this time
5
^{2} + 12
^{2} = 13
^{2}
3
^{2} × 5
^{2} + 3
^{2}× 12
^{2} = 3
^{2}× 13
^{2}
( 3 × 5)
^{2} + ( 3 × 12)
^{2} = ( 3 × 13)
^{2}
15
^{2} + 36
^{2} = 39
^{2}
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
^{2} + 5
^{2} = 7
^{2} ?
4
^{2} + 5
^{2} = 16 + 25 = 41. However, 7
^{2} = 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} + (m
^{2} - 1)
^{2} = (m
^{2} + 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} + (m
^{2} - 1)
^{2} = (m
^{2} + 1)
^{2}
(2× 2)
^{2} + (2
^{2} - 1)
^{2} = (2
^{2} + 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} + (m
^{2} - 1)
^{2} = (m
^{2} + 1)
^{2}
(2× 5)
^{2} + (5
^{2} - 1)
^{2} = (5
^{2} + 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 26
^{2} = 26 × 26 = 676