Which primes divide $x^2-5$?

Question

Which primes divide $x^2-5$?

What I have tried:

If $p$ divides $x^2 -5 $ then: $$x^2= 5\pmod{p}$$ Therefore, from Euler's extended theorem we get that for primes s.t $\gcd(5,p)=1$ (which are all except $5$ which can be checked manually), we have: $$5^{(p-1)/2}=1\pmod{p}$$ These are the primes that $5$ is not a primitive root modulo for.

How can I carry on from here? Are there any other ways to solve this neatly?

Thanks!

Answer 1

By quadratic reciprocity, and since $5\equiv 1\pmod 4$, $5$ is a square mod $p$ (an odd prime) iff $p$ is a square mod $5$, that is, if $p$ is $0$, $1$ or $4$ mod $5$, or more visually, if $p=5$ or the last digit of $p$ is $1$ or $9$.

Of course, $2$ can also divide $x^2-5$, namely when $x$ is odd.