Determine whether $x^2\equiv 2$ mod $59$ has solutions.

Question

Determine whether $x^2\equiv 2$ mod $59$ has solutions.

I know Euler's Criterion.

So I want to determine if $2^{58/2}=2^{29}\equiv 1$ mod $59$

However besides computing this with a calculator, I'm not sure how to determine that this is a quadratic residue mod $59$.

Is there a way to do this which has computations that are easier to do by hand?

Answer 1

No, it has no solution since the Legendre symbol is given by $$ \biggl(\frac{2}{59}\biggr)=(-1)^{(59^2-1)/8}=-1^{435}=-1. $$ Actually, we do not need to compute $(p^2-1)/8$, it is enough to see that $p=59\equiv 3\bmod 8$.