Curious congruences for cyclotomic 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: $2{\Phi }_n^{(3)}(1)$ is divisible by $\varphi (n)-2$ and ${\Phi }_n^{(2k+1)}(1)$ is divisible by $\varphi (n)-2k$ for $k\ge 2$. The congruence stems from a general property of self-reciprocal polynomials.

See remarks we recieved afterwards in the appendix. To appear in Research in Number Theory