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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 $E$."},"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":"1508.02464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","cross_cats_sorted":[],"title_canon_sha256":"5d01891d1c69a559d82b7a6d5ebd92782ad709513a13a5970b466d1f5ba80e99","abstract_canon_sha256":"5a2c6294673f682429e2ae233c3e107a5d2be7494627e0bd10fc8eb41753f8f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:35.123860Z","signature_b64":"yZWfBzAOt4bJFDn3jZDDvBfcKWmC+5wuJQUeDdAlxWJx9cY9ukKJElKN0Ph60aVZ9B1pJL+g6Eb831QWJ5JeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","last_reissued_at":"2026-05-18T00:25:35.123124Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:35.123124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\frac{179}{336}$. 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