Constrained ternary integers

TL;DR

This paper counts ternary integers composed of three distinct odd primes and explores their properties, applying results to cyclotomic polynomials and verifying a conjecture for most such integers.

Contribution

It provides an asymptotic count of ternary integers under various prime constraints and applies these findings to cyclotomic polynomial coefficients, confirming a conjecture for most cases.

Findings

Asymptotic count of ternary integers with prime constraints

Application to cyclotomic polynomial coefficient analysis

Verification of the Sister Beiter conjecture for ≥92.5% of ternary integers

Abstract

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.