Quadratic residues and related permutations and identities

TL;DR

This paper explores quadratic residues modulo an odd prime, analyzing related permutations, identities, and congruences, and connects these findings to class numbers of imaginary quadratic fields, providing new mathematical identities and results.

Contribution

It introduces new identities involving quadratic residues, evaluates complex products using class number formulas, and links permutation signs to class numbers for primes congruent to 3 or 7 mod 8.

Findings

Permutation sign depends on class number and prime congruence

Evaluates complex cotangent product using class number formula

Determines exact cosine product values for quadratic forms

Abstract

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ (with $\{k\}_p$ the least nonnegative residue of $k$ modulo $p$) is a permutation of $a_1,\ldots,a_{(p-1)/2}$, and we show that the sign of this permutation is $1$ or $(-1)^{(h(-p)+1)/2}$ according as $p\equiv3\pmod 8$ or $p\equiv7\pmod 8$, where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. To achieve this, we evaluate the product $\prod_{1\le j<k\le(p-1)/2}(\cot\pi j^2/p-\cot\pi k^2/p)$ via Dirichlet's class number formula and Galois theory. We also obtain some new identities for the sine and cosine functions; for example, we determine the exact value of $$\prod_{1\le j<k\le p-1}\cos\pi\frac{aj^2+bjk+ck^2}p$$ for any $a,b,c\in\mathbb Z$ with $ac(a+b+c)\not\equiv0\pmod p$.