A special family of ternary cyclotomic polynomials

Abstract

Let $A(n)$ denote the largest absolute value of the coefficients of the $n$-th cyclotomic polynomial ${\Phi }_n(x)$. In this paper, for odd primes $7<q<r$ with $r\equiv \pm 3(\mathrm{mod}7q)$, we show that $A(7qr)=1$ if $q\equiv 1(\mathrm{mod}21)$, $2$ if $q\equiv 2,10,11,19,20(\mathrm{mod}21)$, $3$ if $q\equiv 16(\mathrm{mod}21)$, $2$ or $3$ if $q\equiv 4,5,8,13,17(\mathrm{mod}21)$.