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Furthermore, we will prove the set of all tuples in a cycle for $\\mathbb{Z}_m"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2403.05319","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-03-08T13:49:42Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"27bc5f9d7aca8a920b9c94822f547ddce31bede6556a437221a3e156603d10e3","abstract_canon_sha256":"8ed7df5e0038299694425313f8ac609eef1cee0337ff5c1c89723d14ed5a072c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:00:36.410293Z","signature_b64":"gIxo0BPunMtfxCx+HzofSB8kyd5tARrHVbVWuhskEhWnW1eAFdmXaI+N6aRJTYPj0cgL9WSidjLeW0B/ZS+AAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8350e2d69dbd4b96194e25c676f2d2f1ef6331026aa0e0e78fb9c0a705ea853","last_reissued_at":"2026-07-05T09:00:36.409837Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:00:36.409837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ducci on $\\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd","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":"2024-03-08T13:49:42Z","abstract_excerpt":"Define the Ducci function $D: \\mathbb{Z}_m^n \\to \\mathbb{Z}_m^n$ 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).\\]\n  Call $\\{D^{\\alpha}(\\mathbf{u})\\}_{\\alpha=0}^{\\infty}$ the Ducci sequence of $\\mathbf{u}$. Because $\\mathbb{Z}_m^n$ is finite, every Ducci sequence will enter a cycle. In this paper, we will prove that if $n$ is odd and $m=2^lm_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence to enter its cycle is $l$ iterations. 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