Bin Zhang: Remarks on the maximum gap in binary cyclotomic polynomials, p.109-115

Abstract:

Let p < q be odd primes, and let $g(\Phi_{pq})$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in the pq-th cyclotomic polynomial $\Phi_{pq}(x)$. In this note, we give an alternative proof of the following result which was established by Hong et al. [J. Number Theory, 132 (2012), pp. 2297-2315]

$$g(\Phi_{pq})=p-1.$$

In addition, we show that the number of maximum gaps in $\Phi_{pq}(x)$ is given by $2\lfloor\frac{q}{p}\rfloor$.

Key Words: Cyclotomic polynomial, Binary cyclotomic polynomial, Maximum gap.

2000 Mathematics Subject Classification: Primary: 11C08;
Secondary: 11B83..