Partitions of primes by Chebyshev polynomials

TL;DR

This paper introduces a novel way to partition primes using Chebyshev polynomials at rationals, establishing their densities and connecting algebraic and number-theoretic properties through Euler's Criterion, with applications to dynamical systems.

Contribution

It develops a new framework for partitioning primes via Chebyshev polynomials, linking algebraic structures with classical number theory and revealing hidden symmetries.

Findings

Prime densities of partitions are established.

Euler's Criterion for $SL(2,\

Abstract

Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between the algebra of Chebyshev polynomials and number-theoretic properties of the partitions. It is shown how to obtain in this way some of the classical theory of Lucas sequences. A hidden symmetry of the problem is revealed by the new language. As an application number-theoretic properties of simple dynamical systems (rotations and certain interval maps) are discussed.