Example 2 Determine whether the following congruence equations have solutions, and find the number of solutions if they do:
(i) $x^{2} \equiv -1(\bmod 365)$;
(ii) $x^{2} \equiv 2(\bmod 3599)$.
(i) 365 is not a prime, $365=5 \cdot 73$. Therefore, the congruence equation and the system of congruence equations
$$x^{2} \equiv -1(\bmod 5), \quad x^{2} \equiv -1(\bmod 73)$$

are equivalent. By Theorem 1(v), we know
$$\left(\frac{-1}{5}\right)=\left(\frac{-1}{73}\right)=1,$$

so the system of congruence equations has a solution. By Theorem 1 of §4 and Theorem 1 of §5, we know that the original congruence equation has a solution, and the number of solutions is 4.
(ii) 3599 is not a prime, $3599=59 \cdot 61$. The congruence equation is equivalent to the system of congruence equations
$$x^{2} \equiv 2(\bmod 59), \quad x^{2} \equiv 2(\bmod 61)$$

By Theorem 3, we know $\left(\frac{2}{59}\right)=-1$, so there is no solution.