{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:5DVKBNUST2BUG5ULZUMAZH2JMT","short_pith_number":"pith:5DVKBNUS","schema_version":"1.0","canonical_sha256":"e8eaa0b6929e8343768bcd180c9f4964e17914e382a29042f888dac3e3d9d8cc","source":{"kind":"arxiv","id":"1406.5684","version":1},"attestation_state":"computed","paper":{"title":"Perfect Numbers and Fibonacci Primes (II)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang, Tianxin Cai, Yong Zhang","submitted_at":"2014-06-22T08:43:28Z","abstract_excerpt":"In this paper, we study the diophantine equation ${{\\sigma }_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are $n={{F}_{2k+1}}{{F}_{2k+2m+1}}$, where both ${{F}_{2k+1}}$ and ${{F}_{2k+2m+1}}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${{\\sigma }_{2}}(n)-{{n}^{2}}=2n+5$ has infinitely many solutions."},"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":"1406.5684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-22T08:43:28Z","cross_cats_sorted":[],"title_canon_sha256":"1eb3333f59ffb1b1acc30587cc7528594875287e5b130ceba3e559b85ecd3425","abstract_canon_sha256":"b9515d91549333801433ec6057f172069d83f0bb4f606a974792ce9f1d12a7ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:10.981266Z","signature_b64":"7rxQG67o5XkZSWYVSJqjRMZoA5Fp5wg7bpKFok8wj9SliKZbhXdPgdwnmhKRu5v7ok7CND3+ikQW2S0r7mwACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8eaa0b6929e8343768bcd180c9f4964e17914e382a29042f888dac3e3d9d8cc","last_reissued_at":"2026-05-18T02:49:10.980705Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:10.980705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perfect Numbers and Fibonacci Primes (II)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang, Tianxin Cai, Yong Zhang","submitted_at":"2014-06-22T08:43:28Z","abstract_excerpt":"In this paper, we study the diophantine equation ${{\\sigma }_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are $n={{F}_{2k+1}}{{F}_{2k+2m+1}}$, where both ${{F}_{2k+1}}$ and ${{F}_{2k+2m+1}}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${{\\sigma }_{2}}(n)-{{n}^{2}}=2n+5$ has infinitely many solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5684","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1406.5684","created_at":"2026-05-18T02:49:10.980789+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.5684v1","created_at":"2026-05-18T02:49:10.980789+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5684","created_at":"2026-05-18T02:49:10.980789+00:00"},{"alias_kind":"pith_short_12","alias_value":"5DVKBNUST2BU","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"5DVKBNUST2BUG5UL","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"5DVKBNUS","created_at":"2026-05-18T12:28:14.216126+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT","json":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT.json","graph_json":"https://pith.science/api/pith-number/5DVKBNUST2BUG5ULZUMAZH2JMT/graph.json","events_json":"https://pith.science/api/pith-number/5DVKBNUST2BUG5ULZUMAZH2JMT/events.json","paper":"https://pith.science/paper/5DVKBNUS"},"agent_actions":{"view_html":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT","download_json":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT.json","view_paper":"https://pith.science/paper/5DVKBNUS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.5684&json=true","fetch_graph":"https://pith.science/api/pith-number/5DVKBNUST2BUG5ULZUMAZH2JMT/graph.json","fetch_events":"https://pith.science/api/pith-number/5DVKBNUST2BUG5ULZUMAZH2JMT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT/action/storage_attestation","attest_author":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT/action/author_attestation","sign_citation":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT/action/citation_signature","submit_replication":"https://pith.science/pith/5DVKBNUST2BUG5ULZUMAZH2JMT/action/replication_record"}},"created_at":"2026-05-18T02:49:10.980789+00:00","updated_at":"2026-05-18T02:49:10.980789+00:00"}