$x^2 \equiv -2,2 \pmod {122}$

Question

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?

Answer 1

Let's work with the congruence $x^2 \equiv -2 \pmod {122}$.

The ring $\mathbb Z_{122}$ is isomorphic to $\mathbb Z_{2} \times \mathbb Z_{61}$ by the Chinese Remainder Theorem, as you pointed out. Now applying the isomorphism to both sides, you are looking for a solution to $(x,x')^2 = (-2,-2)$ where $x \in \mathbb Z_2$ and $x' \in \mathbb Z_{61}$.

If $\left(-2\over 61\right)=1$ then there are exactly two solutions to $x^2 \equiv -2 \pmod{61}$. The reason for that is because $\mathbb Z_{61}$ is a field, and the factor theorem: The polynomial $X^2+2$ has a unique factorisation $(X-a)(X-b)$ where each of $a$ and $b$ is a root of the polynomial. And the roots must be distinct because if $a$ is a root of that polynomial then so is $-a$, and the only element of a field that is the negation of itself is $0$ (unless the field has characteristic $2$).
On the other hand, there is only one solution to $(x')^2 \equiv -2 \pmod 2$. So multiplying the number of possible values for $x$ and $x'$ gives $2$.

With the congruence $x^2 \equiv 2 \pmod {122}$, use the distributivity of the Legendre symbols. The Legendre symbol of $-1$ is easily shown to be $-1$ in $\mathbb Z_{61}$, showing that no solution is possible.