Let $p$ be an odd prime.

(i) Define the Legendre symbol $\left(\frac{x}{p}\right)$, and show that when $(x, p)=1$, then $\left(\frac{x^{-1}}{p}\right)=\left(\frac{x}{p}\right)$.

(ii) State and prove Gauss's lemma, and use it to evaluate $\left(\frac{-1}{p}\right)$. [You may assume Euler's criterion.]

(iii) Prove that

$$\sum_{x=1}^{p}\left(\frac{x}{p}\right)=0$$

and deduce that

$$\sum_{x=1}^{p}\left(\frac{x(x+1)}{p}\right)=-1$$

Hence or otherwise determine the number of pairs of consecutive integers $z, z+1$ such that $1 \leqslant z, z+1 \leqslant p-1$ and both $z$ and $z+1$ are quadratic residues $\bmod p$.