Pythagorean Triples
Before showing how to generate Pythagorean Triples, let us lay down a definition.
The definition comes right from the Pythagorean Theorem which states that for all integers a, b, and c, c
2= a
2 + b
2
A Pythagorean triple is a set of three whole numbers a,b, and c bigger than zero 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!
How find Pythagorean triples
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 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.
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 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's formula for Pythagorean Triples:
Plato, a Greek Philosopher, came up with a great formula for finding 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
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May 26, 22 06:50 AM
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