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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 -

    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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