8.
Let
\(p\) be a prime of the form
\(p=2q+1\text{,}\) where
\(q\) is prime (recall that
\(q\) is called a Germain prime in this case). Show that
every residue from 1 to
\(p-2\) is either a primitive root of
\(p\) or a quadratic residue. (Hint: Use
Euler’s Criterion, and ask yourself how many possible orders an element of
\(U_p\) can have.)
9.
Prove: if \(p\equiv 3\) (mod \(4\)), and if \(a\not\equiv \pm 1,0\text{,}\) then \(a\) is a QR modulo \(p\) if and only if \(p-a\) is not a QR.
10.
Prove that for any prime \(p\text{,}\) if \(1<i,j<\frac{p}{2}\) and \(i\neq j\text{,}\) then \(i^2\not\equiv j^2\) (mod \(p\)). (Hint: factor!)
11.
Verify the previous exercise for \(p=23\text{.}\)