{"paper":{"title":"Values of Ducci Periods for Sequences on $\\mathbb{Z}_m^n$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Mark L. Lewis, Shannon M. Tefft","submitted_at":"2025-02-05T16:40:05Z","abstract_excerpt":"Let $D: \\mathbb{Z}_m^n \\to \\mathbb{Z}_m^n$ be defined so that \\[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).\\] We call $D$ the Ducci function and the sequence $\\{D^{\\alpha}(\\mathbf{u})\\}_{\\alpha=0}^{\\infty}$ the Ducci sequence of $\\mathbf{u}$ for $\\mathbf{u} \\in \\mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\\text{Per}(\\mathbf{u})$ be the number of tuples in the Ducci cycle of $\\mathbf{u}$, or the period of $\\mathbf{u}$. In this paper, we will look at what different possible values of $\\text{Per}(\\mathbf{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.03348","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.03348/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}