Maximum gap in cyclotomic polynomials

[摘要] 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 (Phi(p1p2) ) = p1-1 for primes p(2) > p(1). In 2017, based on numerous calculations, the following generalization was conjectured: g (Phi mp) = phi(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 Phi mp, which are divisible by the m-th inverse cyclotomic polynomial Psi(m) = xm-1. (c) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Cyclotomic polynomials;Inverse cyclotomic polynomials;Maximum gap