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Thread: 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
    Global Moderator

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    Hint: $\displaystyle x^2\equiv y^2 ~ (p^2) \implies (x-y)(x+y) \equiv 0 ~ (p^2)$
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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 $\displaystyle x>0$, and $\displaystyle x \not\equiv 0 \mod p$, then there exist $\displaystyle k \ge 0$ and $\displaystyle p>r>0$ such that:

    $\displaystyle x=kp+r$

    Also:

    $\displaystyle -x=(-k)p-r=(-1-k)p+(p-r)$

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

    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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