Coefficients of Unitary Cyclotomic Polynomials of Order Three

2021/11/17 by Gennady Bachman, Bachman, Gennady
Computer Science · Mathematics · #11B83 (Primary) 11C08 (Secondary) #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.08847

openalex publication_date 2021/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A unitary cyclotomic polynomial of order three is a polynomial of the form Φ^*_PQR(x)=\frac(x^PQR-1)(x^P-1)(x^Q-1)(x^R-1)(x^PQ-1)(x^QR-1)(x^RP-1)(x-1), where P, Q and R are powers of three distinct primes p, q and r. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of P=p^a, Q=q^b and R=r^c. We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of ℤ.

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