{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:RFBEKMWJ2UWPXOFYGMSFBM3K3N","short_pith_number":"pith:RFBEKMWJ","canonical_record":{"source":{"id":"1901.01063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-04T11:45:46Z","cross_cats_sorted":[],"title_canon_sha256":"3a21d8dfbc5c59cc3ff284cf51dc4f9a4478a0608825b9715e7d3343e17442e8","abstract_canon_sha256":"8c2e466b72beecb013655538c5213dbec89d8b36d59f39d0a496ceb9d47c2f8a"},"schema_version":"1.0"},"canonical_sha256":"89424532c9d52cfbb8b8332450b36adb43a513fa542a247ee710a523049f77fd","source":{"kind":"arxiv","id":"1901.01063","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01063","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01063v1","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01063","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"pith_short_12","alias_value":"RFBEKMWJ2UWP","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RFBEKMWJ2UWPXOFY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RFBEKMWJ","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:RFBEKMWJ2UWPXOFYGMSFBM3K3N","target":"record","payload":{"canonical_record":{"source":{"id":"1901.01063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-04T11:45:46Z","cross_cats_sorted":[],"title_canon_sha256":"3a21d8dfbc5c59cc3ff284cf51dc4f9a4478a0608825b9715e7d3343e17442e8","abstract_canon_sha256":"8c2e466b72beecb013655538c5213dbec89d8b36d59f39d0a496ceb9d47c2f8a"},"schema_version":"1.0"},"canonical_sha256":"89424532c9d52cfbb8b8332450b36adb43a513fa542a247ee710a523049f77fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:57.233560Z","signature_b64":"CeJVLnPmVAWvtPwvBTCxq8yrO9qJNh8N6EvwGsHhtPjXFQ0EHfGwpRWftsHwqfXz689NjJ9FWVvvWgn8oO/ECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89424532c9d52cfbb8b8332450b36adb43a513fa542a247ee710a523049f77fd","last_reissued_at":"2026-05-17T23:56:57.232918Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:57.232918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.01063","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Shf3kL04Pc450ZVDOf/Y92GgMF2fjYnmPi1ED3OYlJJQnDiEhu2xYWhXTpHjtlUJiZxc8xzZ9oq7jOUAghuhCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:17:01.588658Z"},"content_sha256":"bb9c37ff1025d0bed84235eeecc80c6262a8fdc961563ccb9c158a89d65c29f1","schema_version":"1.0","event_id":"sha256:bb9c37ff1025d0bed84235eeecc80c6262a8fdc961563ccb9c158a89d65c29f1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:RFBEKMWJ2UWPXOFYGMSFBM3K3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On members of Lucas sequences which are products of factorials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Mark Sias, Shanta Laishram","submitted_at":"2019-01-04T11:45:46Z","abstract_excerpt":"Here, we show that if $\\{U_n\\}_{n\\ge 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=m_1!m_2!\\cdots m_k!$ with $1<m_1\\le m_2\\le \\cdots\\le m_k$ satisfies $n<3\\times 10^5$. We also give better bounds in case the roots of the Lucas sequence are real."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01063","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G0BlkgWejj3UkcS+N08Ti51UqwBJTu2dieMpEwIUju6karCRtxN7pGch+zrqVXsdmQLCjLDh9wr+8j2QkxrwBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:17:01.589037Z"},"content_sha256":"37ba0b7107784fa9d162ab0e7eb3506af458ce5c5ac998480c20818d52ba851f","schema_version":"1.0","event_id":"sha256:37ba0b7107784fa9d162ab0e7eb3506af458ce5c5ac998480c20818d52ba851f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/bundle.json","state_url":"https://pith.science/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/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":"2026-05-25T21:17:01Z","links":{"resolver":"https://pith.science/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N","bundle":"https://pith.science/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/bundle.json","state":"https://pith.science/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RFBEKMWJ2UWPXOFYGMSFBM3K3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RFBEKMWJ2UWPXOFYGMSFBM3K3N","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":"8c2e466b72beecb013655538c5213dbec89d8b36d59f39d0a496ceb9d47c2f8a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-04T11:45:46Z","title_canon_sha256":"3a21d8dfbc5c59cc3ff284cf51dc4f9a4478a0608825b9715e7d3343e17442e8"},"schema_version":"1.0","source":{"id":"1901.01063","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01063","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01063v1","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01063","created_at":"2026-05-17T23:56:57Z"},{"alias_kind":"pith_short_12","alias_value":"RFBEKMWJ2UWP","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RFBEKMWJ2UWPXOFY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RFBEKMWJ","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:37ba0b7107784fa9d162ab0e7eb3506af458ce5c5ac998480c20818d52ba851f","target":"graph","created_at":"2026-05-17T23:56:57Z","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":"Here, we show that if $\\{U_n\\}_{n\\ge 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=m_1!m_2!\\cdots m_k!$ with $1<m_1\\le m_2\\le \\cdots\\le m_k$ satisfies $n<3\\times 10^5$. We also give better bounds in case the roots of the Lucas sequence are real.","authors_text":"Florian Luca, Mark Sias, Shanta Laishram","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-04T11:45:46Z","title":"On members of Lucas sequences which are products of factorials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01063","kind":"arxiv","version":1},"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:bb9c37ff1025d0bed84235eeecc80c6262a8fdc961563ccb9c158a89d65c29f1","target":"record","created_at":"2026-05-17T23:56:57Z","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":"8c2e466b72beecb013655538c5213dbec89d8b36d59f39d0a496ceb9d47c2f8a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-04T11:45:46Z","title_canon_sha256":"3a21d8dfbc5c59cc3ff284cf51dc4f9a4478a0608825b9715e7d3343e17442e8"},"schema_version":"1.0","source":{"id":"1901.01063","kind":"arxiv","version":1}},"canonical_sha256":"89424532c9d52cfbb8b8332450b36adb43a513fa542a247ee710a523049f77fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89424532c9d52cfbb8b8332450b36adb43a513fa542a247ee710a523049f77fd","first_computed_at":"2026-05-17T23:56:57.232918Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:57.232918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CeJVLnPmVAWvtPwvBTCxq8yrO9qJNh8N6EvwGsHhtPjXFQ0EHfGwpRWftsHwqfXz689NjJ9FWVvvWgn8oO/ECQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:57.233560Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.01063","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb9c37ff1025d0bed84235eeecc80c6262a8fdc961563ccb9c158a89d65c29f1","sha256:37ba0b7107784fa9d162ab0e7eb3506af458ce5c5ac998480c20818d52ba851f"],"state_sha256":"1d35ae45fc48581c3cb9e36965761b2329e7d8b31766251cf14ffc2f7ddd5fd3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BP+47GpK+1UPvSytIfjxCLlXszZHXNv3+2hsPOpo9/GbjjfCoLgNtGMbYXUWsSsajVN4X1ff5oBXWrO/AxMXDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T21:17:01.591307Z","bundle_sha256":"cb51662c54b3aeef564f64e02fca2643104525893aa3fd79ad72c3ed0b899c91"}}