Legendre symbol (39/p)

Question

$$\left(\frac{39}{p}\right)=\left(\frac{3}{p}\right)\left(\frac{13}{p}\right)$$

$$\left(\frac{3}{p}\right) is\;easy: $$

$$\begin{align} \left(\frac{3}{p}\right) = \begin{Bmatrix} 1 & \mathbf{if} \quad p \equiv 1 \pmod {12} \quad \mathbf{or} \quad p \equiv 11 \pmod {12}\\ -1 & \mathbf{if} \quad p \equiv 5 \pmod {12} \quad \mathbf{or} \quad p \equiv 7 \pmod {12}\\ \end{Bmatrix} \end{align}$$

$$And\;what\;about\;\left(\frac{13}{p}\right)\;?$$

Help please

Answer 1

We can use Euler's Criterion to calculate a Legendre Symbol:

$$\left(\frac ap\right)=\left(a^{\frac{p-1}{2}}\mod p\right)\in\{-1,0,1\}$$

Therefore, we have $$\left(\frac {13}p\right)=\left(13^{\frac{p-1}{2}}\mod p\right)$$

Does this help in any way?