Divisibility criteria and coefficient formulas for cyclotomic polynomials

2025/08/01 by De Carli, Laura, Laporta, Maurizio
#12D10 05E05 11C08 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.02722

Abstract

We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the coefficients of cyclotomic polynomials and we propose a recursive relation between the coefficients of two cyclotomic polynomials whose indexes differ by a prime factor.

Citations

• A Survey on Coefficients of Cyclotomic Polynomials
• Combinatorial and harmonic-analytic methods for integer tilings
• Coefficients and higher order derivatives of cyclotomic polynomials: old\n and new
• Coefficients of cyclotomic polynomials
• The spectral set conjecture and multiplicative properties of roots of polynomials
• Tiling the integers with translates of one finite set
• The Midterm Coefficient of the Cyclotomic Polynomial F pq (x)

Related

• A Survey on Coefficients of Cyclotomic Polynomials
• Some properties of coefficients of cyclotomic polynomials
• Coefficients of cyclotomic polynomials
• Unitary Cyclotomic Polynomials
• Block Structure of Cyclotomic Polynomials