# Proof of some conjectures involving quadratic residues

## TL;DR

This paper proves several conjectures by Sun related to quadratic residues modulo odd primes, involving products, residue comparisons, and properties of quadratic fields, advancing understanding of quadratic residue patterns and their algebraic connections.

## Contribution

It provides the first proofs of Sun's conjectures on quadratic residues, connecting residue counts with class numbers and fundamental units of quadratic fields.

## Key findings

- Confirmed conjectures involving quadratic residue products and residue comparisons.
- Established relationships between residue counts and class numbers of quadratic fields.
- Extended results to primes congruent to 3 mod 4 involving triangular numbers.

## 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 $m(m+1)/2$.