The flatness of ternary cyclotomic polynomials (Q2039362)

Summary: 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 \(zr\equiv\pm\pmod{pq}\), where \(z\) is a positive integer. So far, the classification of flat ternary cyclotomic polynomials for \(1\leq z\leq 5\) has been given. In this paper, for \(z=6\) and \(q\equiv\pm 1\pmod{p}\), we give the necessary and sufficient conditions for ternary cyclotomic polynomials \(\Phi_{pqr}(x)\) to be flat.