Maximum gap in cyclotomic polynomials

Abstract

We study the maximum gap g (maximum of the differences between any two consecutive exponents) of cyclotomic polynomials. In 2012, Hong, Lee, Lee and Park showed that g(Φp1p2)=p1−1 for primes p2>p1. In 2017, based on numerous calculations, the following generalization was conjectured: g(Φmp)=φ(m) for square free odd m and prime p>m. The main contribution of this paper is a proof of this conjecture. The proof is based on the discovery of an elegant structure among certain sub-polynomials of Φmp, which are divisible by the m-th inverse cyclotomic polynomial Ψm=[Formula presented]. © 2021 Elsevier Inc.