I've read about this formula on wikipedia, but attempting to use it just gets me:
$$q\equiv 3 \bmod 4\implies p\equiv 1 \mod 4 $$ and$$q\equiv 1 \bmod 4\implies p\equiv 1,3 \mod 4.$$ However, $$1,4,9,3,12,10\equiv x^2\bmod 13 19$$ disproves this. I also don't see how derive $\pm2$ implying $1$ or $7 \mod 8.$
What am I missing,how does Legendre symbol formula $(-1)^{{p-1\over 2}{q-1\over 2}}$, actually show the correct value ?
I do partially understand Euler's criterion.
According to the law of quadratic reciprocity, $$\left(\dfrac pq\right)\left(\dfrac qp\right)=(-1)^{\dfrac{p-1}2\dfrac{q-1}2},$$ where $\left(\dfrac pq\right)=1$ if $n^2\equiv p \pmod q$ for some $n$ and $-1$ otherwise.
For example, $1319\equiv3\pmod4$. $3$ is a quadratic residue modulo $1319$,
but $1319\equiv2\pmod3$ is not a quadratic residue modulo $3$.
In this case the quadratic reciprocity law equality is $-1=-1$.