{"paper":{"title":"Complete Padovan sequences in finite fields","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Catalin Zara, Juan B. Gil, Michael D. Weiner","submitted_at":"2006-05-12T19:37:58Z","abstract_excerpt":"Given a prime $p\\ge 5$, and given $1<\\kappa<p-1$, we call a sequence $(a_n)_{n}$ in $\\mathbb{F}_p$ a $\\Phi_{\\kappa}$-sequence if it is periodic with period $p-1$, and if it satisfies the linear recurrence $a_n+a_{n+1}=a_{n+\\kappa}$ with $a_0=1$. Such a sequence is said to be a complete $\\Phi_{\\kappa}$-sequence if in addition $\\{a_0,a_1,...,a_{p-2}\\}=\\{1,...,p-1\\}$. For instance, every primitive root $b$ mod $p$ generates a complete $\\Phi_{\\kappa}$-sequence $a_n=b^n$ for some (unique) $\\kappa$. A natural question is whether every complete $\\Phi_{\\kappa}$-sequence is necessarily defined by a pri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0605348","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":""},"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"}