The flatness of ternary cyclotomic polynomials

Abstract

It is well known that all of the prime cyclotomic polynomials and binary cyclotomic polynomials are flat, and the flatness of ternary cyclotomic polynomials is much more complicated. Let $p<q<r$ be odd primes such that $zr\equiv \pm$ (mod $pq$), where $z$ is a positive integer. So far, the classification of flat ternary cyclotomic polynomials for $1\le z\le 5$ has been given. In this paper, for $z=6$ and $q\equiv \pm 1$ (mod $p$), we give the necessary and sufficient conditions for ternary cyclotomic polynomials ${\Phi }_{pqr}(x)$ to be flat.