Non-Beiter ternary cyclotomic polynomials with an optimally large set of coefficients

TL;DR

This paper constructs infinitely many ternary cyclotomic polynomials with maximal coefficient sets that contradict the Beiter conjecture, expanding understanding of coefficient bounds in cyclotomic polynomials.

Contribution

It demonstrates the existence of infinitely many such polynomials with large coefficient sets that violate the Beiter conjecture, providing new counterexamples.

Findings

Existence of infinitely many such polynomials.

Coefficient sets reach the known maximum size.

Contradicts the Beiter conjecture for these polynomials.

Abstract

Let l>=1 be an arbitrary odd integer and p,q and r primes. We show that there exist infinitely many ternary cyclotomic polynomials \Phi_{pqr}(x) with l^2+3l+5<= p<q<r such that the set of coefficients of each of them consists of the p integers in the interval [-(p-l-2)/2,(p+l+2)/2]. It is known that no larger coefficient range is possible. The Beiter conjecture states that the cyclotomic coefficients a_{pqr}(k) of \Phi_{pqr} satisfy |a_{pqr}(k)|<= (p+1)/2 and thus the above family contradicts the Beiter conjecture. The two already known families of ternary cyclotomic polynomials with an optimally large set of coefficients (found by G. Bachman) satisfy the Beiter conjecture.