{"paper":{"title":"Using Lucas Sequences to Generalize a Theorem of Sierpi\\'nski","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lenny Jones","submitted_at":"2011-06-10T12:11:41Z","abstract_excerpt":"In 1960, Sierpi\\'nski proved that there exist infinitely many odd positive integers $k$ such that $k\\cdot 2^n+1$ is composite for all positive integers $n$. In this paper, we prove some generalizations of Sierpi\\'nski's theorem with $2^n$ replaced by expressions involving certain Lucas sequences $U_n(\\alpha,\\beta)$. In particular, we show the existence of infinitely many Lucas pairs $(\\alpha,\\beta)$, for which there exist infinitely many positive integers $k$, such that $k (U_n(\\alpha,\\beta)+(\\alpha-\\beta)^2)+1$ is composite for all integers $n\\ge 1$. Sierpi\\'nski's theorem is the special case"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2029","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"}