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    Difference of two squares

    Prove that a positive even integer is a difference of two squares if and only if it is divisible by 4.
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    Quote Originally Posted by tarheelborn View Post
    Prove that a positive even integer is a difference of two squares if and only if it is divisible by 4.
    WLOG, let a > b \geq 0, k = a^2 - b^2, k \text{ even}.

    From considerations modulo 2, a and b must both be even, or both be odd.

    We can rewrite k = (a + b)(a - b)

    Both factors are even.

    Thus 4 divides k.

    Sorry, forgot about the "if" and only did the "only if." Hmm.

    Okay, for the other part, let n=4k > 0

    It is not hard to show that n must have two factors with an even difference. Just find two factors of k, as in k=cd and take n=(2c)(2d).

    So we can take the midpoint of the two even factors, which would allow us to write n=(a+b)(a-b).
    Last edited by undefined; April 22nd 2010 at 10:22 AM.
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