Balanced paring of $\{1,2,\ldots,(p-1)/2\}$ for $p\equiv 1 \pmod{4}$

TL;DR

This paper constructs a balanced pairing of elements in the set {1, 2, ..., (p-1)/2} for primes p ≡ 1 mod 4, and uses this to address a problem related to quadratic residues and permutation signs.

Contribution

It introduces a novel balanced pairing method for quadratic residues modulo p and applies it to solve a problem posed by Zhi-Wei Sun.

Findings

The pairing is balanced with equal frequencies of three order cases.

The method provides insights into the structure of quadratic residues.

It resolves a specific problem related to permutation signs and quadratic residues.

Abstract

Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$. Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $t a \equiv \pm \tilde{a} \pmod{p}.$ For any two such pairs, assume $a<\tilde{a}, b<\tilde{b}, a<b $, then there are three possibilities for their relative order : $a<\tilde{a} < b< \tilde{b}$ , $a< b < \tilde{a} < \tilde{b}$ , $a< b < \tilde{b}< \tilde{a}$. We show this paring is balanced in the sense that the three cases occur with equal frequencies. Utilizing properties of this paring we solve one problem raised by Zhi-Wei Sun concerning the sign of permutation related to quadratic residues.