Some conjectures on the maximal height of divisors of $x^n-1$

Abstract

Define $B\left(n\right)$ to be the largest height of a polynomial in $\mathbb{Z}\left[x\right]$ dividing $x^n-1$. We formulate a number of conjectures related to the value of $B\left(n\right)$ when $n$ is of a prescribed form. Additionally, we prove a lower bound for $B\left(n\right)$.