Some properties of coefficients of cyclotomic polynomials

Abstract

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with $t\geq 3$ odd, then the numbers $-(r-2),-(r-3),\ldots, r-2, r-1$ are all coefficients of the cyclotomic polynomial $\Phi_{2n}$. Furthermore, if $1+p_r<p_1+p_2$ then $1-r$ is also a coefficient of $\Phi_{2n}$.} In the experimental part, in two instances we present computational evidence for asymptotic symmetry between distribution of positive and negative coefficients, and state the resulting conjectures.}