# The flatness of ternary cyclotomic polynomials

### Bin Zhang

Qufu Normal University, China

## 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 $1\leq z\leq 5$ 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.

## Cite this article

Bin Zhang, The flatness of ternary cyclotomic polynomials. Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 1–42

DOI 10.4171/RSMUP/47