(i) State and prove the Fermat-Euler Theorem.

(ii) Let $p$ be an odd prime number, and $x$ an integer coprime to $p$. Show that $x^{(p-1) / 2} \equiv \pm 1(\bmod p)$, and that if the congruence $y^{2} \equiv x(\bmod p)$ has a solution then $x^{(p-1) / 2} \equiv 1(\bmod p)$.

(iii) By arranging the residue classes coprime to $p$ into pairs $\{a, b x\}$ with $a b \equiv 1(\bmod p)$, or otherwise, show that if the congruence $y^{2} \equiv x(\bmod p)$ has no solution then $x^{(p-1) / 2} \equiv-1(\bmod p) .$

(iv) Show that $5^{5^{5}} \equiv 5(\bmod 23)$.