{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HLFF4KN7NTXZUNCCYMEHEBD75E","short_pith_number":"pith:HLFF4KN7","canonical_record":{"source":{"id":"1310.0898","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-03T05:08:44Z","cross_cats_sorted":[],"title_canon_sha256":"fb5ad5ff0c9e3b1c5ee08b45d96d6de378a93fbaff61eb936bf2c83a2d9f0932","abstract_canon_sha256":"e7c8e321c27327f803095ffdfb4a654194032c0df1d48797d22e1cce673f583c"},"schema_version":"1.0"},"canonical_sha256":"3aca5e29bf6cef9a3442c30872047fe93d74fdfd7c08ced1caf0e420cccafa37","source":{"kind":"arxiv","id":"1310.0898","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0898","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0898v2","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0898","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"pith_short_12","alias_value":"HLFF4KN7NTXZ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HLFF4KN7NTXZUNCC","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HLFF4KN7","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HLFF4KN7NTXZUNCCYMEHEBD75E","target":"record","payload":{"canonical_record":{"source":{"id":"1310.0898","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-03T05:08:44Z","cross_cats_sorted":[],"title_canon_sha256":"fb5ad5ff0c9e3b1c5ee08b45d96d6de378a93fbaff61eb936bf2c83a2d9f0932","abstract_canon_sha256":"e7c8e321c27327f803095ffdfb4a654194032c0df1d48797d22e1cce673f583c"},"schema_version":"1.0"},"canonical_sha256":"3aca5e29bf6cef9a3442c30872047fe93d74fdfd7c08ced1caf0e420cccafa37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:00.775447Z","signature_b64":"Ri95M8Dzx4vGSLd5eOVZCKxylKZZT/1dZ4Nk2Qw/iraVQ7MmPVXQJ8QDf0ejIsM0ALKHTdxHkxWc41orFUluCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aca5e29bf6cef9a3442c30872047fe93d74fdfd7c08ced1caf0e420cccafa37","last_reissued_at":"2026-05-18T02:50:00.774945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:00.774945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.0898","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:50:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OHx39YCPifSrt7ODNYri5Tj1lhAWawE8u+5ox0KInB49KHLhZvXYjh7I5XUw7qK74/wjZFkcT6Uhinr/Fjw9CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T15:51:40.726553Z"},"content_sha256":"4fe1a7429cb55cc7f7f40ddaaacf362dd55e4f74a92e8dde440dcec7884c4468","schema_version":"1.0","event_id":"sha256:4fe1a7429cb55cc7f7f40ddaaacf362dd55e4f74a92e8dde440dcec7884c4468"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HLFF4KN7NTXZUNCCYMEHEBD75E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perfect numbers and Fibonacci primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Deyi Chen, Tianxin Cai, Yong Zhang","submitted_at":"2013-10-03T05:08:44Z","abstract_excerpt":"In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0898","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:50:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"atxbRW+d9sVaRtAo09qcAB8PfX7Xk4fMqKC3lge46w3TK43GkN5/u53ktDJLDzrjIS7CwR7xQrkMczwjEh+yBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T15:51:40.726878Z"},"content_sha256":"1c6d3a2370bca5202c4df690fb23131b25ddc62fc7337b4b1089b8e13246fa91","schema_version":"1.0","event_id":"sha256:1c6d3a2370bca5202c4df690fb23131b25ddc62fc7337b4b1089b8e13246fa91"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/bundle.json","state_url":"https://pith.science/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T15:51:40Z","links":{"resolver":"https://pith.science/pith/HLFF4KN7NTXZUNCCYMEHEBD75E","bundle":"https://pith.science/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/bundle.json","state":"https://pith.science/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HLFF4KN7NTXZUNCCYMEHEBD75E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HLFF4KN7NTXZUNCCYMEHEBD75E","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e7c8e321c27327f803095ffdfb4a654194032c0df1d48797d22e1cce673f583c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-03T05:08:44Z","title_canon_sha256":"fb5ad5ff0c9e3b1c5ee08b45d96d6de378a93fbaff61eb936bf2c83a2d9f0932"},"schema_version":"1.0","source":{"id":"1310.0898","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0898","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0898v2","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0898","created_at":"2026-05-18T02:50:00Z"},{"alias_kind":"pith_short_12","alias_value":"HLFF4KN7NTXZ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HLFF4KN7NTXZUNCC","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HLFF4KN7","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:1c6d3a2370bca5202c4df690fb23131b25ddc62fc7337b4b1089b8e13246fa91","target":"graph","created_at":"2026-05-18T02:50:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.","authors_text":"Deyi Chen, Tianxin Cai, Yong Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-03T05:08:44Z","title":"Perfect numbers and Fibonacci primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0898","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4fe1a7429cb55cc7f7f40ddaaacf362dd55e4f74a92e8dde440dcec7884c4468","target":"record","created_at":"2026-05-18T02:50:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e7c8e321c27327f803095ffdfb4a654194032c0df1d48797d22e1cce673f583c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-03T05:08:44Z","title_canon_sha256":"fb5ad5ff0c9e3b1c5ee08b45d96d6de378a93fbaff61eb936bf2c83a2d9f0932"},"schema_version":"1.0","source":{"id":"1310.0898","kind":"arxiv","version":2}},"canonical_sha256":"3aca5e29bf6cef9a3442c30872047fe93d74fdfd7c08ced1caf0e420cccafa37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3aca5e29bf6cef9a3442c30872047fe93d74fdfd7c08ced1caf0e420cccafa37","first_computed_at":"2026-05-18T02:50:00.774945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:00.774945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ri95M8Dzx4vGSLd5eOVZCKxylKZZT/1dZ4Nk2Qw/iraVQ7MmPVXQJ8QDf0ejIsM0ALKHTdxHkxWc41orFUluCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:00.775447Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.0898","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4fe1a7429cb55cc7f7f40ddaaacf362dd55e4f74a92e8dde440dcec7884c4468","sha256:1c6d3a2370bca5202c4df690fb23131b25ddc62fc7337b4b1089b8e13246fa91"],"state_sha256":"a9ea6b5e2ac2fadb61423bfc725743c0fec004a57f21cfca84477b8fe4251758"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tRVatirTMQIxkfgQXuepTEpnGJ4Ma1dUBEZ8jAzImrAQoMXBqbuwZb9J/RrdzUWg3ULOUy/VmYfXGioXtg4qDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T15:51:40.728938Z","bundle_sha256":"7e893b062c24480866d0b8259952d695aade2af041ea1583ded637b5888a8079"}}