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

Abstract

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