Find all solutions (if there are any) to the following congruences

Question

Find all solutions (if there are any):

$x^2 \equiv -1 \pmod{7}$

I found $x$ $\equiv$ $\sqrt6 \pmod{7}$

Is this the answer?

Can I use the same procedure for $x^2 \equiv -1 \pmod{13}$

Answer 1

You haven't described your procedure, so how should we be able to answer that?

Modular arithmetic is normally only interesting in $\mathbb N\not\ni \sqrt{6}$, so chances are that you haven't found what was searched for, which would suggest you need a different approach.

Answer 2

I think the question requires you to find integer solutions, and there are none.

In fact, there are no solutions to $x^{2} \equiv -1 \pmod p$ if $p$ is prime and satisfies $p \equiv 3 \pmod 4$.

$$x^2 \equiv -1 \pmod{p} \implies x^4 \equiv 1 \pmod{p}$$

We have $p = 4m+3$ which gives us

$$x^{p-1} = x^{4m+2} = x^2x^{4m} \equiv-1\pmod{p}$$

This would be a contradiction to Fermat's little theorem, and hence has no solutions, for $x \in \mathbb{Z}$.