let $N$ be odd and $N-1 = 2^t u$ with u being odd. Let $X_N$ be the set of all elements that pass the Miller-Rabin-Test, i.e.
$X_N = \{ x \in \mathbb{Z}_N^\times | x^u \equiv 1 \mod N \text{ or } \exists s \in \{ 0,\dots,t-1 \}: x^{2^s u} \equiv -1 \mod N \}$.
My problem is it to determine $|X_N|$ for certain numbers $N$, let's say $N=2993$ or $N=6859$. Our exercise instructor wants us to use the fact that
- there exists a primitive root modulo $N$ iff N is 2,4 or a power of a prime $p$ with $p\neq2$ and
- that $x^2 \equiv 1 \mod N \Longleftrightarrow x \in \{-1,1\} \mod N$ in case of $N$ being prime.
But I do not understand how we should use these hints. We are also not allowed to compute them rigorously, meaning with an Brute-Force approach.
Could someone help me with this problem? Thanks a lot.
