A property of cyclotomic polynomials

TL;DR

This paper investigates the linear combinations of two distinct cyclotomic polynomials, focusing on the minimal natural number that can be expressed as such a combination with integer polynomial coefficients.

Contribution

It determines the minimal natural number k for which a linear combination of two different cyclotomic polynomials equals k, advancing understanding of their algebraic properties.

Findings

Identifies the minimal k for linear combinations of two cyclotomic polynomials.

Provides explicit conditions for representing integers as combinations of cyclotomic polynomials.

Enhances knowledge of the algebraic structure of cyclotomic polynomials.

Abstract

Given two cyclotomic polynomials $\Phi_n(x)$ and $\Phi_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\Phi_n (x)+b(x)\Phi_m(x),$$ with a(x) and b(x) integer polynomials.