On a generalization of Beiter Conjecture

TL;DR

This paper demonstrates that for any small positive epsilon and fixed integer omega, there exist primes such that the height of the corresponding cyclotomic polynomial exceeds a certain bound, generalizing Beiter's conjecture.

Contribution

It provides a construction showing the height of cyclotomic polynomials can be arbitrarily large, extending the Beiter conjecture to a broader class of polynomials.

Findings

Existence of primes with large cyclotomic polynomial height

Explicit lower bounds involving product of primes

Asymptotic behavior of the constant c_omega

Abstract

We prove that for every $\varepsilon>0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\ldots,p_\omega$ such that for $n=p_1p_2\ldots p_\omega$ the height of the cyclotomic polynomial $\Phi_n$ is at least $(1-\varepsilon)c_\omega M_n$, where $M_n=\prod_{i=1}^{\omega-2}p_i^{2^{\omega-1-i}-1}$ and $c_\omega$ is a constant depending only on $\omega$; furthermore $\lim_{\omega\to\infty}c_\omega^{2^{-\omega}}\approx0.71$. In our construction we can have $p_i>h(p_1p_2\ldots p_{i-1})$ for all $i=1,2,\ldots,\omega$ and any function $h:\mathbb{R}_+\to\mathbb{R}_+$.