Establish the following assertion: (5/p)=1 if and only if p=1, 9, 11, or 19(mod 20)
$\displaystyle \left ( \frac{5}{p} \right ) = \left ( \frac{p}{5} \right ) (-1)^{(p-1)(5-1)/4} = \left ( \frac{p}{5} \right )$
Therefore $\displaystyle p \equiv 1, 4 \pmod{5}$
Therefore $\displaystyle p \equiv 1, 4, 6,9, 11,14,16,19 \pmod{20}$
However, p is odd, therefore
$\displaystyle p \equiv 1, 9, 11, 19 \pmod{20}$