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Math Help - Proof Method

  1. #1
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    Proof Method

    Hi all,

    I need the proof of the following:
    If A, B , and C are positive integers such that C^2= A^2 + B^2 then either A or B is an even integer.

    I could be grateful if you help me.

    Raed
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  2. #2
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    If A and B are both odd, then C^2 is a multiple of 4. (It can't be even but not divisible by 4, right?) Then if you represent A and B as 2 times an even number plus 1, then the right-hand side is not divisible by 4.
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  3. #3
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    Quote Originally Posted by raed View Post
    Hi all,

    I need the proof of the following:
    If A, B , and C are positive integers such that C^2= A^2 + B^2 then either A or B is an even integer.

    I could be grateful if you help me.

    Raed
    You could also try to prove that the squares of a pair of odd integers do not sum to an integer square...

    A=2f+1,\;\;\;B=2j+1

    C will have to be even, since Odd^2+Odd^2=Even

    Odd squares are Odd and Even squares are Even, hence C is Even.

    C=2k

    A^2+B^2=(2f+1)^2+(2g+1)^2=4f^2+4f+1+4g^2+4g+1

    This needs to equal 4k^2

    \Rightarrow\ 2\left(2f^2+2g^2+2f+2g+1\right)=2\left(2k^2\right)

    2\left(f^2+g^2+f+g\right)+1=2k^2

    However, this is contradictory as the LHS is Odd.
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