{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:DBG4BWWGKFENZUERNPDSMC7MQJ","short_pith_number":"pith:DBG4BWWG","canonical_record":{"source":{"id":"0907.3412","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-20T14:51:34Z","cross_cats_sorted":[],"title_canon_sha256":"bcdca85d38ae456677d676690b59fd48b00a0fa5e41405559fb66ca65f5b5674","abstract_canon_sha256":"4ab981694ffcc8dda8175c92bb84286b489876012764feccb9445ed15d257d54"},"schema_version":"1.0"},"canonical_sha256":"184dc0dac65148dcd0916bc7260bec824d6f0627292308ba30a9e42267be95a0","source":{"kind":"arxiv","id":"0907.3412","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3412","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3412v2","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3412","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"pith_short_12","alias_value":"DBG4BWWGKFEN","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"DBG4BWWGKFENZUER","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"DBG4BWWG","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:DBG4BWWGKFENZUERNPDSMC7MQJ","target":"record","payload":{"canonical_record":{"source":{"id":"0907.3412","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-20T14:51:34Z","cross_cats_sorted":[],"title_canon_sha256":"bcdca85d38ae456677d676690b59fd48b00a0fa5e41405559fb66ca65f5b5674","abstract_canon_sha256":"4ab981694ffcc8dda8175c92bb84286b489876012764feccb9445ed15d257d54"},"schema_version":"1.0"},"canonical_sha256":"184dc0dac65148dcd0916bc7260bec824d6f0627292308ba30a9e42267be95a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:51.739277Z","signature_b64":"Nx99w0v+jsvkHVURoSLbUcgtI5PbH3CLkxvn6bn9GrNDujL9kjDiboN1sjmq5L5wqqU0zHGQo+s2QWXxWAdRCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"184dc0dac65148dcd0916bc7260bec824d6f0627292308ba30a9e42267be95a0","last_reissued_at":"2026-05-18T03:52:51.738344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:51.738344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.3412","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":"2026-05-18T03:52:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OeLU0kUtVqO3r5enlkslb+Pp5bBBklnKpMGr26lhzoDtn3ME8x4/LT7nOv5Nv5ESgMs5ZNs1Hsaq24bw70F2Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:49:19.928873Z"},"content_sha256":"50777607e0a25b57416b46403ba0e0b403d948b925ecafa11cd3c4b1b99a6307","schema_version":"1.0","event_id":"sha256:50777607e0a25b57416b46403ba0e0b403d948b925ecafa11cd3c4b1b99a6307"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:DBG4BWWGKFENZUERNPDSMC7MQJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The 2-adic valuations of Stirling numbers of the second kind","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianrong Zhao, Shaofang Hong, Wei Zhao","submitted_at":"2009-07-20T14:51:34Z","abstract_excerpt":"In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\\not\\equiv 7\\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3412","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":"2026-05-18T03:52:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EC6vaJi8s8wnFQq07SbrLSn6Ah74j+Brliu43AdrH2tCS8iN6H0LIki4TzG5RJhsHJqxWBdrlBVSdtEmSOIYDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:49:19.929209Z"},"content_sha256":"1dcc92f458f45e3c2cdceaa317d6e2cbdabc74b5ca23bcc9445cdc5f1568d4eb","schema_version":"1.0","event_id":"sha256:1dcc92f458f45e3c2cdceaa317d6e2cbdabc74b5ca23bcc9445cdc5f1568d4eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/bundle.json","state_url":"https://pith.science/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/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":"2026-06-10T21:49:19Z","links":{"resolver":"https://pith.science/pith/DBG4BWWGKFENZUERNPDSMC7MQJ","bundle":"https://pith.science/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/bundle.json","state":"https://pith.science/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DBG4BWWGKFENZUERNPDSMC7MQJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:DBG4BWWGKFENZUERNPDSMC7MQJ","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":"4ab981694ffcc8dda8175c92bb84286b489876012764feccb9445ed15d257d54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-20T14:51:34Z","title_canon_sha256":"bcdca85d38ae456677d676690b59fd48b00a0fa5e41405559fb66ca65f5b5674"},"schema_version":"1.0","source":{"id":"0907.3412","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3412","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3412v2","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3412","created_at":"2026-05-18T03:52:51Z"},{"alias_kind":"pith_short_12","alias_value":"DBG4BWWGKFEN","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"DBG4BWWGKFENZUER","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"DBG4BWWG","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:1dcc92f458f45e3c2cdceaa317d6e2cbdabc74b5ca23bcc9445cdc5f1568d4eb","target":"graph","created_at":"2026-05-18T03:52:51Z","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 investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\\not\\equiv 7\\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.","authors_text":"Jianrong Zhao, Shaofang Hong, Wei Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-20T14:51:34Z","title":"The 2-adic valuations of Stirling numbers of the second kind"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3412","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":"sha256:50777607e0a25b57416b46403ba0e0b403d948b925ecafa11cd3c4b1b99a6307","target":"record","created_at":"2026-05-18T03:52:51Z","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":"4ab981694ffcc8dda8175c92bb84286b489876012764feccb9445ed15d257d54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-20T14:51:34Z","title_canon_sha256":"bcdca85d38ae456677d676690b59fd48b00a0fa5e41405559fb66ca65f5b5674"},"schema_version":"1.0","source":{"id":"0907.3412","kind":"arxiv","version":2}},"canonical_sha256":"184dc0dac65148dcd0916bc7260bec824d6f0627292308ba30a9e42267be95a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"184dc0dac65148dcd0916bc7260bec824d6f0627292308ba30a9e42267be95a0","first_computed_at":"2026-05-18T03:52:51.738344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:51.738344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nx99w0v+jsvkHVURoSLbUcgtI5PbH3CLkxvn6bn9GrNDujL9kjDiboN1sjmq5L5wqqU0zHGQo+s2QWXxWAdRCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:51.739277Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.3412","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50777607e0a25b57416b46403ba0e0b403d948b925ecafa11cd3c4b1b99a6307","sha256:1dcc92f458f45e3c2cdceaa317d6e2cbdabc74b5ca23bcc9445cdc5f1568d4eb"],"state_sha256":"f7b31f75023c477b5b1134e5721ca5ad84234781e2fcb7799675495ad7d66dc9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B/SqUgR8ez6sdRQmz5h2JglE/koTRtytooOzzNiX0+9Z9VcKuwwnhWB4WojZ3iNnNN+kMVkFI7twFmDVjH5/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:49:19.931096Z","bundle_sha256":"bd96fdb51a6cdeaab6cfb98fa5eec45e8426ab9403484bcfa9ca83232b37eee4"}}