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.
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