{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:WDZKAUXMWN2C26LBOUDTBXZUWG","short_pith_number":"pith:WDZKAUXM","canonical_record":{"source":{"id":"1408.5381","kind":"arxiv","version":10},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"496ee05a089a81e1e4400161739631fe4efe53eeb395c6bec553b284656afe27","abstract_canon_sha256":"da4086f8a1e247b3c5f4fa46eea958d599dfcf57a435640b8c83f6745f1aaf60"},"schema_version":"1.0"},"canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","source":{"kind":"arxiv","id":"1408.5381","version":10},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.5381","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"arxiv_version","alias_value":"1408.5381v10","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5381","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"pith_short_12","alias_value":"WDZKAUXMWN2C","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WDZKAUXMWN2C26LB","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WDZKAUXM","created_at":"2026-05-18T12:28:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:WDZKAUXMWN2C26LBOUDTBXZUWG","target":"record","payload":{"canonical_record":{"source":{"id":"1408.5381","kind":"arxiv","version":10},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"496ee05a089a81e1e4400161739631fe4efe53eeb395c6bec553b284656afe27","abstract_canon_sha256":"da4086f8a1e247b3c5f4fa46eea958d599dfcf57a435640b8c83f6745f1aaf60"},"schema_version":"1.0"},"canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:10.936090Z","signature_b64":"Qq0w6TDEgOsy/KB7f9j4rpsOiByZ36Rm22jRICceIgEZi7WO0ehgfIHF0NWmuh5DfVhsyg4HDcmJJsF+cXljCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","last_reissued_at":"2026-05-18T00:01:10.935411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:10.935411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.5381","source_version":10,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E9ONBWrfvYUhgcPOAvtz3EAtHbryJ/ciFAE8sEo+RKRUJmwM9nOEuu5psklcOqQrAd6ysEypukQ8l5iVVRlEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T07:08:55.492949Z"},"content_sha256":"8d0371288e0c0a9b636820165c54facc58f4322372f2662c2f0840917162cba5","schema_version":"1.0","event_id":"sha256:8d0371288e0c0a9b636820165c54facc58f4322372f2662c2f0840917162cba5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:WDZKAUXMWN2C26LBOUDTBXZUWG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two new kinds of numbers and related divisibility results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-08-21T16:15:50Z","abstract_excerpt":"We mainly introduce two new kinds of numbers given by $$R_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k\\frac1{2k-1}\\quad\\ (n=0,1,2,...)$$ and $$S_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k(2k+1)\\quad\\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\\equiv1\\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\\equiv1\\pmod 4$ and $y\\equiv0\\pmod 2$), then $$R_{(p-1)/2}\\equiv p-(-1)^{(p-1)/4}2x\\pmod{p^2}.$$ Also, $$\\frac1{n^2}\\sum_{k=0}^{n-1}S_k\\in\\mathbb Z\\ \\ {and}\\ \\ \\frac1n\\sum_{k=0}^{n-1}S_k(x)\\in\\mathbb Z[x]\\quad\\text{for all}\\ n=1,2,3,...,$$ where $S_k(x)=\\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5381","kind":"arxiv","version":10},"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":"2026-05-18T00:01:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uiTi1hn+T42VxLVFhxVRfEqTmWeCoDdvuOApFxRTwIC2FG8cTYhhAcMefxAoPYN8TI109bEKmV2SCUN4IgxCDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T07:08:55.493640Z"},"content_sha256":"ff65c8e5c1fd21fc466e267892e5f3adb6e70dff8af51b3121b0c32a577ce77c","schema_version":"1.0","event_id":"sha256:ff65c8e5c1fd21fc466e267892e5f3adb6e70dff8af51b3121b0c32a577ce77c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/bundle.json","state_url":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/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":"2026-05-24T07:08:55Z","links":{"resolver":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG","bundle":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/bundle.json","state":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WDZKAUXMWN2C26LBOUDTBXZUWG","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":"da4086f8a1e247b3c5f4fa46eea958d599dfcf57a435640b8c83f6745f1aaf60","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","title_canon_sha256":"496ee05a089a81e1e4400161739631fe4efe53eeb395c6bec553b284656afe27"},"schema_version":"1.0","source":{"id":"1408.5381","kind":"arxiv","version":10}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.5381","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"arxiv_version","alias_value":"1408.5381v10","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5381","created_at":"2026-05-18T00:01:10Z"},{"alias_kind":"pith_short_12","alias_value":"WDZKAUXMWN2C","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WDZKAUXMWN2C26LB","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WDZKAUXM","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:ff65c8e5c1fd21fc466e267892e5f3adb6e70dff8af51b3121b0c32a577ce77c","target":"graph","created_at":"2026-05-18T00:01:10Z","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":"We mainly introduce two new kinds of numbers given by $$R_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k\\frac1{2k-1}\\quad\\ (n=0,1,2,...)$$ and $$S_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k(2k+1)\\quad\\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\\equiv1\\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\\equiv1\\pmod 4$ and $y\\equiv0\\pmod 2$), then $$R_{(p-1)/2}\\equiv p-(-1)^{(p-1)/4}2x\\pmod{p^2}.$$ Also, $$\\frac1{n^2}\\sum_{k=0}^{n-1}S_k\\in\\mathbb Z\\ \\ {and}\\ \\ \\frac1n\\sum_{k=0}^{n-1}S_k(x)\\in\\mathbb Z[x]\\quad\\text{for all}\\ n=1,2,3,...,$$ where $S_k(x)=\\s","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","title":"Two new kinds of numbers and related divisibility results"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5381","kind":"arxiv","version":10},"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":"sha256:8d0371288e0c0a9b636820165c54facc58f4322372f2662c2f0840917162cba5","target":"record","created_at":"2026-05-18T00:01:10Z","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":"da4086f8a1e247b3c5f4fa46eea958d599dfcf57a435640b8c83f6745f1aaf60","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","title_canon_sha256":"496ee05a089a81e1e4400161739631fe4efe53eeb395c6bec553b284656afe27"},"schema_version":"1.0","source":{"id":"1408.5381","kind":"arxiv","version":10}},"canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","first_computed_at":"2026-05-18T00:01:10.935411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:10.935411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qq0w6TDEgOsy/KB7f9j4rpsOiByZ36Rm22jRICceIgEZi7WO0ehgfIHF0NWmuh5DfVhsyg4HDcmJJsF+cXljCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:10.936090Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.5381","source_kind":"arxiv","source_version":10}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d0371288e0c0a9b636820165c54facc58f4322372f2662c2f0840917162cba5","sha256:ff65c8e5c1fd21fc466e267892e5f3adb6e70dff8af51b3121b0c32a577ce77c"],"state_sha256":"266cf032b437393cd9afc16551f00af74ded82fbbc2ce2481bf248243e337842"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qUOokqL7I8zcjcGpQcKGK6+rZrGgG2WfWajvqOCyKcnOdtu90XGcD36MQhoMqKYmEI0AgC29lnexjR41+dPGDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T07:08:55.497204Z","bundle_sha256":"1da640d1aa8b4dda0a6216d43873fd4e2ec080f3b5fddc26c7b1b7259295fd1d"}}