Maximum gap in cyclotomic polynomials

2019/11/26 by Al-Kateeb, Ala'a, Ambrosino, Mary, Hong, Hoon +1 · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.11667

Abstract

Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap g (maximum of the differences between any two consecutive exponents). In 2012, it was shown that g( Φ_p₁p₂) =p₁ -1 for primes p₂>p₁. In 2017, based on numerous calculations, the following generalization was conjectured: g( Φ_mp) =φ(m) for square free odd m and prime p>m. The main contribution of this paper is a proof of this conjecture.

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