Polynomials in Base $x$ and the Prime-Irreducible Affinity

TL;DR

This paper explores the relationship between polynomials in base $x$, prime numbers, and irreducibility, revealing a unique base-$x$ representation and a correspondence between primes and certain irreducible polynomials, with implications for number theory conjectures.

Contribution

It introduces a unique base-$x$ representation of polynomials and establishes a correspondence between primes and irreducible polynomials generated from cyclotomic polynomials.

Findings

Existence of a unique base-$x$ representation making $ ext{Z}[x]$ an ordered domain.

A one-to-one correspondence between primes and certain irreducible polynomials.

Potential implications for the Bouniakowsky Conjecture.

Abstract

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique "base-$x$" representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.