Posted by **Kevin** on Wednesday, February 22, 2012 at 6:24am.

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.

- Math -
**MathMate**, Wednesday, February 22, 2012 at 7:43am
Verification:

5²=25=12+13

5²+12²=25+144=169=13²

Therefore it works.

Proof:

Let the odd number be a=2k+1,

where k=any integer.

(2k+1)²=4k²+4k+1

The two consecutive numbers are therefore

b=2k²+2k and c=2k²+2k+1

Now calculate the pythagorean sum of a and b:

a²+b²

=(2k+1)²+(2k²+2k)²

=4k^4+8k^3+8k²+4k+1

=(2k²+2k+1)²

=c²

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