Posted by **Kevin** on Wednesday, February 22, 2012 at 6:27am.

How to Generate Pythagorean Triples using an Odd Number

1. Start with an odd number, e.g. 3. Then the square of 3 can be expressed as the sum of two consecutive numbers, i.e. 32 = 9 = 4 + 5. Therefore the Pythagorean Triples are {3, 4, 5} since 32 + 42 = 52. Generate another set of Pythagorean Triples starting with the next odd number 5 and verify that it works.

2. Generate another set of Pythagorean Triples starting with the next odd number 7 and verify that it works.

3. Prove that this method will always work for any odd number except 1.

- Math -
**Damon**, Wednesday, February 22, 2012 at 7:22am
please write exponent with ^ symbol

3^2 = 9

5, ?

25 = sum of 12 and 13

25 + 144 = 169 ?

yes

so 5, 12, 13 works

x^2 = n + (n+1)

is x^2 + n^2 = (n+1)^2 ???

x^2 + n^2 = n^2 +2 n + 1 ???

so if x^2 = 2 n + 1 we have it

but we see 2 n + 1 = n + (n+1)

so true

but what if n is even

try x = 2

x^2 = 4

BUT 4, an even number can not be made from the sum of two consecutive whole numbers. So x must be odd.

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