{"paper":{"title":"The Period of Ducci Cycles on $\\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$","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":"2024-01-30T23:29:21Z","abstract_excerpt":"Let the Ducci function $D: \\mathbb{Z}_m^n \\to \\mathbb{Z}_m^n$ be defined as\n  \\[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)\\]\n  and let the Ducci sequence of $\\mathbf{u}$ be the sequence $\\{D^{\\alpha}(\\mathbf{u})\\}_{\\alpha=0}^{\\infty}$.\n  %In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen.\n  In this paper, we will provide another proof that for $n=2^k$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.17502","kind":"arxiv","version":2},"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/2401.17502/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"}