Let $p=2n+1$ be an odd prime, and let $a_1<\ldots<a_{n}$ be all the quadratic residues mod $p$ among $1,\ldots,p-1$. For $a\in\mathbb Z$ let $\{a\}_p$ be the least nonnegative residue of $a$ modulo $p$. It is well known that the list $\{1^2\}_p,\ldots,\{n^2\}_p$ is a permutation of $a_1,\ldots,a_n$. Recently, I determined the sign of this permutation for $p\equiv3\pmod4$ in the preprint arXiv:1809.07766 (available from http://arxiv.org/abs/1809.07766). Motivated by this, here I ask the following question.

QUESTION. Is it true that for each prime $p\equiv3\pmod4$ we have \begin{align*}&\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \text{and}\ \{j(j+1)\}_p> \{k(k+1)\}_p\right\}\right| \\&\quad\qquad\equiv\left\lfloor\frac{p+1}8\right\rfloor\pmod 2\ ? \end{align*}

I have verified this for all primes $p<20000$ with $p\equiv 3\pmod4$. Based on this, I conjecture that the question has a positive answer. Any ideas towards its solution?

Let $p=2n+1$ be an odd prime, and let $a_1<\ldots<a_{n}$ be all the quadratic residues mod $p$ among $1,\ldots,p-1$. For $a\in\mathbb Z$ let $\{a\}_p$ be the least nonnegative residue of $a$ modulo $p$. It is well known that the list $\{1^2\}_p,\ldots,\{n^2\}_p$ is a permutation of $a_1,\ldots,a_n$. Recently, I determined the sign of this permutation for $p\equiv3\pmod4$ in the preprint arXiv:1809.07766 (available from http://arxiv.org/abs/1809.07766). Motivated by this, here I ask the following question.

QUESTION. Is it true that for each prime $p\equiv3\pmod4$ we have \begin{align*}&\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \text{and}\ \{j(j+1)\}_p> \{k(k+1)\}_p\right\}\right| \\&\quad\qquad\equiv\left\lfloor\frac{p+1}8\right\rfloor\pmod 2\ ? \end{align*}

I have verified this for all primes $p<20000$ with $p\equiv 3\pmod4$. Based on this, I conjecture that the question has a positive answer. Any ideas towards its solution?