Curious congruences for cyclotomic polynomials

2022/04/24 by Shigeki Akiyama, Akiyama, Shigeki, Hajime Kaneko +1
Mathematics · #11A07 #11A25 #11R18 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.11267

openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Φ_n^(k)(x) be the k-th derivative of n-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: 2Φ⁽³⁾_n(1) is divisible by ϕ(n)-2 and Φ^(2k+1)_n(1) is divisible by ϕ(n)-2k for k≥ 2. The congruence stems from a general property of self-reciprocal polynomials.

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