We introduce the Möbius polynomial 
, which gives the number of aperiodic bracelets of length n with x possible types of gems, and therefore satisfies Mn(x) ≡ 0 for all (mod n) for all x ϵ 핫. 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.
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