Curious congruences for cyclotomic polynomials

TL;DR

This paper explores intriguing divisibility properties of derivatives of cyclotomic polynomials at 1, extending Lehmer's work and revealing new congruences related to Euler's totient function.

Contribution

It establishes new divisibility congruences for derivatives of cyclotomic polynomials at 1, generalizing previous results and linking them to properties of self-reciprocal polynomials.

Findings

$2 ext{Phi}^{(3)}_n(1)$ divisible by $(n)-2$

$ ext{Phi}^{(2k+1)}_n(1)$ divisible by $(n)-2k$ for $k extgreater 1$

Derived from properties of self-reciprocal polynomials

Abstract

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