{"paper":{"title":"The density of primes dividing a particular non-linear recurrence sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexi Block Gorman, Heesu Hwang, Jeremy Rouse, Noam Kantor, Sarah Parsons, Tyler Genao","submitted_at":"2015-08-11T01:24:16Z","abstract_excerpt":"Define the sequence $\\{b_n\\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\\begin{cases} \\frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\not\\equiv 0\\pmod 3, \\frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\equiv 0\\pmod 3. We relate this sequence $\\{b_n\\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02464","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":""},"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"}