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Math Help - Square Root Proof Problem

  1. #1
    Newbie cryptocrow's Avatar
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    Square Root Proof Problem

    Hi. I'm having trouble with these problems.

    p is and odd prime for both.

    Prove x = 0 mod p if x = -x mod p.

    Prove x = +/-y mod p^2 if x^2 = y^2 mod p^2, neither x nor y are 0 mod p.

    Thanks...
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  2. #2
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    Hint: x^2\equiv y^2 ~ (p^2) \implies (x-y)(x+y) \equiv 0 ~ (p^2)
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by cryptocrow View Post
    Hi. I'm having trouble with these problems.

    p is and odd prime for both.

    Prove x = 0 mod p if x = -x mod p.

    Prove x = +/-y mod p^2 if x^2 = y^2 mod p^2, neither x nor y are 0 mod p.

    Thanks...
    Suppose x>0, and x \not\equiv 0 \mod p, then there exist k \ge 0 and p>r>0 such that:

    x=kp+r

    Also:

    -x=(-k)p-r=(-1-k)p+(p-r)

    So if x \equiv -x \mod p then p-r=r, or 2r=p, but p is an odd prime which is a contradiction, so our premis fails and x \equiv 0 \mod p.

    RonL
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  4. #4
    Newbie cryptocrow's Avatar
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    Thanks a lot, i see where i went wrong.
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