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Curious congruences for cyclotomic polynomials
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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.
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Cited by 1 Pith paper
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Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products
Odd-order reciprocal power sums satisfy a uniform Bernoulli-polynomial congruence modulo n, and Lehmer-type products admit truncated Bell-polynomial expansions modulo n^{K+1}.
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