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

Nov 15, 18 05:01 PM
Modeling multiplication with number counters  Learning multiplication is fun!
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.