A remark on Zoloterav's theorem

TL;DR

This paper explores the relationship between permutation signs derived from modular arithmetic and the Jacobi symbol, revealing a parity connection depending on the congruence class of n modulo 4.

Contribution

It establishes a new link between permutation signs defined by a specific rule and classical number-theoretic symbols, extending Zoloterav's theorem.

Findings

Sign of gamma_{a,n} matches (a/n) when n ≡ 1 mod 4

Sign of gamma_{a,n} is always 1 when n ≡ 3 mod 4

Provides a new perspective on permutation signs and quadratic residues

Abstract

Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gamma_{a,n} of {1,...,(n-1)/2} by gamma_{a,n}(x)=n-\dec{ax}_n if {ax}_n>=(n+1)/2, and {ax}_n if {ax}_n<=(n-1)/2, where {x}_n denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of gamma_{a,n} coincides with the Jacobi symbol (a/n) if n=1 mod 4, and 1 if n=3 mod 4.