Modified Congruence Modulo $n$ with Half The Amount of Residues

TL;DR

This paper introduces a novel congruence relation that halves the residue classes in modular arithmetic, providing new insights and simplifying the analysis of number theoretic properties, especially for primes and semiprimes.

Contribution

It defines a new congruence relation that reduces residue classes by half, enabling elegant descriptions of existing theorems and applications to primitive roots and minimal polynomials.

Findings

Halves the number of residue classes in modular arithmetic

Simplifies analysis of powers of odd primes and semiprimes

Provides new perspectives on cyclotomic polynomials and primitive roots

Abstract

We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of residue classes. Using it, we are able to give an elegant description of some results due to Carl Schick, others are reduced to well-known theorems from algebra and number theory. Many concepts from number theory such as quadratic residues and primitive roots are equally applicable. It brings noticeable advantages in studying powers of odd primes, and in particular when studying semiprimes composed of a pair of related primes, e.g. a pair of twin primes. Artin's primitive root conjecture can be formulated in the new context. Trigonometric polynomials based on chords and related to the new congruence relation lead to new insights into the minimal polynomials of $2\cos(2\pi/n)$ and their relation to cyclotomic polynomials.