Prove that p is a quadratic residue mod $\frac{p^2+3p-2}{2}$

Question

Let p be a prime number. Suppose that $q = \frac{p^2+3p-2}{2}$ is also a prime number. Prove that there is some integer $x$ so that $x^2 \equiv p$ (mod q). I assume I'm supposed to use quadratic reciprocity here, but I have no idea how to go about doing that. Any help would be greatly appreciated.

Answer 1

Observe first $p$ has to be congruent to $1$ mod $4$. Let apart the obvious case $p=2$. If $p\equiv 3\mod4$, then $2q=p^2+3p-2\equiv 1+1-2\equiv 0\mod 4$, whence $q\equiv 0\mod 2$ is not prime.

On the other hand, $\;q\equiv -1\mod p$, and since $p\equiv 1\mod 4$, the first supplementary law of quadratic reciprocity says that $\Bigl(\dfrac{-1}p\Bigr)=1$, so $$\Bigl(\dfrac pq\Bigr)=\Bigl(\dfrac pq\Bigr)\Bigl(\dfrac qp\Bigr)=(-1)^{\tfrac{p-1}2\tfrac{q-1}2}=1.$$