Proof of some conjectures involving quadratic residues (1907.12981v3)

Abstract: We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|{1\le k<\frac p4:\ (\frac kp)=-1}|}\prod_{1\le j<k\le(p-1)/2}(e^{2\pi iaj^2/p}+e^{2\pi iak^2/p}) \\=&\begin{cases}1&\text{if}\ p\equiv1\pmod 8,\\\left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}&\text{if}\ p\equiv5\pmod8,\end{cases} \end{align*} and that \begin{align*}&\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \{aj^2\}_p>{ak^2}_p\right}\right| \&+\left|\left{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \{ak^2-aj^2\}_p>\frac p2\right}\right| \\equiv&\left|\left{1\le k<\frac p4:\ \left(\frac kp\right)=\left(\frac ap\right)\right}\right|\pmod2. \end{align*} where $(\frac{a}p)$ is the Legendre symbol, $\varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $\mathbb Q(\sqrt p)$ respectively, and ${x}_p$ is the least nonnegative residue of an integer $x$ modulo $p$. Also, for any prime $p\equiv3\pmod4$ and $\delta=1,2$, we determine $$(-1)^{\left|\left{(j,k): \ 1\le j<k\le(p-1)/2\ \text{and}\ \{\delta T_j\}_p>{\delta T_k}_p\right}\right|},$$ where $T_m$ denotes the triangular number $m(m+1)/2$.