Examples

Example 3.2   Is $6$ a square modulo $389$? We have

$$\left(\frac{6}{389}\right) = \left(\frac{2\cdot 3}{389}\right) =\left(\frac{2}{389}\right)\cdot \left(\frac{3}{389}\right) = (-1)\cdot(-1)= 1.$$

Here, we found that $\left(\frac{2}{389}\right) = -1$ using Proposition 2.3 and that $389\equiv 3\pmod{8}$. We found $\left(\frac{3}{389}\right)$ as follows:

$$\left(\frac{3}{389}\right) = \left(\frac{389}{3}\right) = \left(\frac{2}{3}\right)=-1.$$

Thus $6$ is a square modulo $389$.

Annoyingly, though we know that $6$ is a square modulo $389$, we still don't know an $x$ such that $x^2\equiv 6\pmod{389}$!

? for(a=1,388,if(Mod(a,389)^2==6,print1(a, " ")))
28 361

Example 3.3   Is $3$ a square modulo $p=726377359$? We proved that the answer is ``no'' in the previous lecture by computing $3^{p-1}\pmod{p}$. It's easier to prove that the answer is no using Theorem 3.1:

$$\left(\frac{3}{726377359}\right) = (-1)^{1\cdot \frac{726377358}{2}}\cdot \left(\frac{726377359}{3}\right) = -\left(\frac{1}{3}\right) = -1.$$