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Pythagorean Triples
Before showing how to generate Pythagorean Triples, let us lay down a definition.
A Pythagorean triple is a set of three whole numbers a,b, and c bigger than zero such that a2 + b2 = c2
Notice that c is listed last and that is very important!
How find Pythagorean triples
Example #132 + 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 and the Pythagorean triple is (6,8,10)
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.
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 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's formula for Pythagorean Triples:
Plato, a Greek Philosopher, came up with a great formula for finding 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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