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Contents
Pythagorean Triples
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Pythagorean Triples
Pythagorean
Triples are a set of three integer numbers that satisfy the Pythagorean
equation.
a2
+ b2 = c2
Examples
of Pythagorean triples are:
a b c
3 4 5
5 12 13
7 24 25
9 40 41
etc.
How
do we generate Pythagorean triples? One way, of course, is trial and error, or
memorising what you already know. However, there is a formula you can use.
We start with the number a. For this to work, a must be an odd
integer. Take the square of a, which is a2. Now, a2
is an odd number. Split it into two components, b and c, where c = b
+ 1.To
highlight this idea, let us use one of the examples above. Take a to be
7. Square 7 and you will get 49. Splitting 49 into two numbers one bigger than
another, you will get 24 and 25. Try this with other odd numbers!
So
far, you know the method for odd numbers. What about even numbers, then?
The Pythagorean triple property for a set of integer numbers will hold when
all three numbers are multiplied by the same number. Take the primitive
Pythagoran triple, 3 4 5, as an example.Multiplying
all the numbers by 2, you will get 6, 8, 10.
Now,
62 + 82 = 36 + 64 = 100 = 102
Multiplying
all the numbers by 3, you will get 9, 12, 15.
Now,
92 + 122 = 81 + 144 = 225 = 152
Hence, to find the Pythagorean triple for the smallest number 2a being
even, find the triple for a, an odd number, and multiply its triple by
2. Note
that to find the triple of an even number, it must have an odd factor. What
about the powers of 2, which comes down to factors 1 and 2? As 1 does not have
a triple, and neither does 2, such numbers do not have a triple as well.
This
formula is not the only way to generate triples. Have fun working with
triples.
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