Example 2 Find an odd prime $p$ such that
$$x^{2} \equiv 5(\bmod p)$$

List the solution
From $5 \equiv 1(\bmod 4)$ and Theorem 2, we know
$$\left(\frac{5}{p}\right)=\left(\frac{p}{5}\right)$$

Thus, when $p \equiv 1$ or $-1(\bmod 5)$, $\left(\frac{p}{5}\right)=1$. In this case, equation (44) has a solution. When $p \equiv 2$ or $-2(\bmod 5)$, $\left(\frac{p}{5}\right)=-1$, in which case equation (44) has no solution.