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Tefft","submitted_at":"2024-10-23T18:16:45Z","abstract_excerpt":"Let $D: \\mathbb{Z}_m^n \\to \\mathbb{Z}_m^n$ be defined so \\[D(x_1, x_2, ..., x_n)=(x_1+x_2 \\; \\text{mod} \\; m, x_2+x_3 \\; \\text{mod} \\; m, ..., x_n+x_1 \\; \\text{mod} \\; m).\\] $D$ is known as the Ducci function and for $\\mathbf{u} \\in \\mathbb{Z}_m^n$, $\\{D^{\\alpha}(\\mathbf{u})\\}_{\\alpha=0}^{\\infty}$ is the Ducci sequence of $\\mathbf{u}$. Every Ducci sequence enters a cycle because $\\mathbb{Z}_m^n$ is finite. 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