Some properties of coefficients of cyclotomic polynomials

TL;DR

This paper explores properties of cyclotomic polynomial coefficients through theoretical proofs and experimental evidence, revealing specific coefficient patterns and conjecturing asymptotic symmetry in their distribution.

Contribution

It provides new theoretical results on coefficients of cyclotomic polynomials and offers experimental evidence for symmetry conjectures.

Findings

Proved specific coefficient sets for certain cyclotomic polynomials.

Presented computational evidence for symmetry in coefficient distribution.

Formulated conjectures based on experimental data.

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.}