{"paper":{"title":"On second order linear sequences of composite numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adrienne Ko, Celine Lee, Dan Ismailescu, Jae Yong Park","submitted_at":"2018-12-19T15:57:58Z","abstract_excerpt":"In this paper we present a new proof of the following 2010 result of Dubickas, Novikas, and Siurys:\n  Let $(a,b)\\in \\mathbb{Z}^2$ and let $(x_n)_{n\\ge 0}$ be the sequence defined by some initial values $x_0$ and $x_1$ and the second order linear recurrence \\begin{equation*} x_{n+1}=ax_n+bx_{n-1} \\end{equation*} for $n\\ge 1$. Suppose that $b\\neq 0$ and $(a,b)\\neq (2,-1), (-2, -1)$. Then there exist two relatively prime positive integers $x_0$, $x_1$ such that $|x_n|$ is a composite integer for all $n\\in \\mathbb{N}$.\n  The above theorem extends a result of Graham who solved the problem when $(a,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08041","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"}