I am trying to solve the following problem:
Which of the following congruences has solutions? How many?
$$x^2 \equiv 2 \pmod {122}$$ $$x^2 \equiv -2 \pmod {122}$$
For both congruences, $122 = 2\times61$. Hence, each congruence can be decomposed to the following: $x^2 \equiv 2 \pmod 2$ and $x^2 \equiv 2 \pmod{61}$. For the first one, $x$ has a unique solution $x = 0$. for the second one, I need to compute $\left(\frac{2}{61}\right)$ which is $-1$.
Now Can I conclude that the congruence is unsolvable? Hence, there exist no solutions?
For the second problem, the congruence $x^2 \equiv -2 \pmod {61}$ is solvable.
Can I conclude that there is one or two solutions?