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Möbius Polynomials

We introduce the Möbius polynomial $ M_n(x) = \sum_{d|n} μ\left( \frac nd \right) x^d $, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \equiv 0$ (mod $n$) for all $x \in \mathbb Z$. We derive some key properties, analyze graphs in the complex plane, and then apply Möbius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.