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Math Help - A congruence problem

  1. #1
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    A congruence problem

    Prove that x^2 = -a^2 mod p does not have a solution for p a prime = 3 mod4 and a = any integer not divisible by p (or prove that it has a solution if and only if p is a prime 1 mod 4). Please don't use the legendre symbol because I haven't learned that yet.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by machack View Post
    Prove that x^2 = -a^2 mod p does not have a solution for p a prime = 3 mod4 and a = any integer not divisible by p (or prove that it has a solution if and only if p is a prime 1 mod 4). Please don't use the legendre symbol because I haven't learned that yet.
    Since  (a,p)=1 this is the same as solving  (xa^{-1})^2\equiv-1\bmod{p} or in other words solving  y^2\equiv-1\mod{p} .

    I know you don't know about Legendre symbols yet, but do you know how to solve this?
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