Gaps in binary cyclotomic polynomials

TL;DR

This paper characterizes the second gap in binary cyclotomic polynomials for odd primes and introduces a new approach using circular maps to analyze their coefficients.

Contribution

It provides a precise formula for the second gap and counts gaps of each length for certain prime congruences, using a novel coefficient analysis method.

Findings

Second gap equals max of r-1 and p-r-1, with r as the remainder of q mod p.

Number of gaps of each length determined for q ≡ ±1 mod p.

Coefficients described as concatenations from iterations of a circular map.

Abstract

For odd prime numbers $p < q$, let $\Phi_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\Phi_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $\Phi_{pq}$ are described as concatenations of words arising from iterations of a circular map.