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Also, we explain when $\\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\\mathscr{A}_{E,k}$ using the M\\\"obius function."},"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":"1708.08357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-28T14:57:41Z","cross_cats_sorted":[],"title_canon_sha256":"50c2b5517ec8ad96f9993e8410d52ce56900e0ec5ca7498fdb4520da602d4c69","abstract_canon_sha256":"049e8ba25865e7c572b109244284039dd13950c4059eece1d5bcef3d23a44495"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:36.297945Z","signature_b64":"eAi/Lmug5szxw5C/pQ00qz8BPvyKGugq1/scyeVz75H5GsPB1BU9TPmEX0gAMOfIDC1tFh/NvT3yiq/lamy8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36ef1455de902b9274e95239f39422e57143505d331325eb74b4c471c0de2018","last_reissued_at":"2026-05-18T00:36:36.297527Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:36.297527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Seoyoung Kim","submitted_at":"2017-08-28T14:57:41Z","abstract_excerpt":"Let $\\mathbf{D}=(D_{n})_{n\\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\\mathscr{A}_{E,k}=\\{n\\geq 1 : \\gcd(n,D_{n})=k\\}$. We give an explicit structural description of $\\mathscr{A}_{E,k}$. Also, we explain when $\\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. 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