On the coefficients of the cyclotomic polynomials of order three

TL;DR

This paper investigates the coefficients of cyclotomic polynomials of order three, providing conditions under which the Corrected Beiter conjecture holds, and proves it for the specific case p=7.

Contribution

It offers a sufficient condition for the Corrected Beiter conjecture and confirms its validity when p=7.

Findings

Proposed a sufficient condition for the Corrected Beiter conjecture.

Proved the conjecture for the case p=7.

Analyzed the behavior of coefficients for cyclotomic polynomials of order three.

Abstract

We say that a cyclotomic polynomial Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter conjectured that A(pqr)<=(p+1)/2. In 2008, Yves Gallot and Pieter Moree showed that the conjecture is false for every p>=11, and they proposed the Corrected Beiter conjecture: M(p)<=2p/3. Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.