
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

