Flat Cyclotomic Polynomials: A New Approach

We build a new theory for analyzing the coefficients of any cyclotomic polynomial by considering it as a gcd of simpler polynomials. Using this theory, we generalize a result known as periodicity and provide two new families of flat cyclotomic polynomials. One, of order 3, was conjectured by Broadhurst: $\Phi_{pqr}(x)$ is flat where $p<q<r$ are primes and there is a positive integer $w$ such that $r\equiv\pm w\pmod{pq}$, $p\equiv1\pmod w$ and $q\equiv1\pmod{wp}$. The other is the first general family of order 4: $\Phi_{pqrs}(x)$ is flat for primes $p,q,r,s$ where $q\equiv-1\pmod p$, $r\equiv\pm1\pmod{pq}$, and $s\equiv\pm1\pmod{pqr}$. Finally, we prove that the natural extension of this second family to order 5 is never flat, suggesting that there are no flat cyclotomic polynomials of order 5.