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From Apostol Chapter $10$ q$6$:

Assume $m>2$, $(a,m)=1$ and there exists an $x$ such that $x^2\equiv a \pmod m$.

Prove that $x^2\equiv a \pmod m$ has exactly two solutions iff $m$ has a primitive root.

I can do the "if" part by explicitly showing that if $g$ is a primitive root of $m$ then if $g^k$ is a soln then $g^{k+\phi(m)/2}$ is the other distinct soln and any soln is congruent to one or the other.

However no progress on the "only if" part.

  • 0
    I still can't see how to put the answer in an analytic number theory form.2011-12-14
  • 1
    Despite the title of Apostol's book, not everything in it is analytic by nature.2011-12-17

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