{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UCGDQJK7MK7A3BR2IP5P7ZNBN7","short_pith_number":"pith:UCGDQJK7","canonical_record":{"source":{"id":"1501.00573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b2cdea41e556fcfb9ba956745e12e5d721bf7bf3a9a58e75bf06d2cf7d8fbcd5","abstract_canon_sha256":"2bdcda7acc289a88853bd0e394b2362f019d48fc0a8daf86484c22c1f524be65"},"schema_version":"1.0"},"canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","source":{"kind":"arxiv","id":"1501.00573","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00573","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00573v1","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00573","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"pith_short_12","alias_value":"UCGDQJK7MK7A","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UCGDQJK7MK7A3BR2","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UCGDQJK7","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UCGDQJK7MK7A3BR2IP5P7ZNBN7","target":"record","payload":{"canonical_record":{"source":{"id":"1501.00573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b2cdea41e556fcfb9ba956745e12e5d721bf7bf3a9a58e75bf06d2cf7d8fbcd5","abstract_canon_sha256":"2bdcda7acc289a88853bd0e394b2362f019d48fc0a8daf86484c22c1f524be65"},"schema_version":"1.0"},"canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:07.211780Z","signature_b64":"bc0kOelAidIaH3r+c31/B2IWosHp3AWrXE9VUb5rgpxPXZATAQOENoQY4fUtKSge76EjnO9C0ehoYHzYenLaAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","last_reissued_at":"2026-05-18T02:30:07.211045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:07.211045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.00573","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":"2026-05-18T02:30:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IHuRFbzJ3cpmI8FVkTurgxGE5uLSjMmyjtUhdtyntf4x/nWbq6KGI/XK///xwLmhdWcE3evFq7+qDsvip/KFBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:47:03.318558Z"},"content_sha256":"339c89c033435e670bb5d23e0b4a774b033fad7936ebd7168b7737bf88a5fac8","schema_version":"1.0","event_id":"sha256:339c89c033435e670bb5d23e0b4a774b033fad7936ebd7168b7737bf88a5fac8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UCGDQJK7MK7A3BR2IP5P7ZNBN7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-01-03T15:54:44Z","abstract_excerpt":"The numbers $R_n$ and $W_n$ are defined as \\begin{align*} R_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{1}{2k-1},\\ \\text{and}\\ W_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{3}{2k-3}. \\end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \\begin{align*} \\sum_{k=0}^{n-1}(2k+1)R_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)R_k^2 &\\equiv 4p(-1)^{\\frac{p-1}{2}} -p^2 \\pmod{p^3}, \\\\ 9\\sum_{k=0}^{n-1}(2k+1)W_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)W_k^2 &\\equiv 12p(-1)^{\\frac{p-1}{2}}-17p^2 \\pmod{p^3}, \\quad\\text{if $p>3$.} \\end{align*} The firs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00573","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":"2026-05-18T02:30:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d5OPkupC51l5MfMhP8nJM+UDbxhv5akZvGPnNXss90F5f/bwCldKgAJA8s2YpuU2/biR+7PjrTzRB+1J/1tFBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:47:03.319179Z"},"content_sha256":"cc0b897cc7f2c3a126cb96bbd10a400a858a76e8f0285daae82025c44f3c8283","schema_version":"1.0","event_id":"sha256:cc0b897cc7f2c3a126cb96bbd10a400a858a76e8f0285daae82025c44f3c8283"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/bundle.json","state_url":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/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":"2026-05-26T02:47:03Z","links":{"resolver":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7","bundle":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/bundle.json","state":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UCGDQJK7MK7A3BR2IP5P7ZNBN7","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":"2bdcda7acc289a88853bd0e394b2362f019d48fc0a8daf86484c22c1f524be65","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","title_canon_sha256":"b2cdea41e556fcfb9ba956745e12e5d721bf7bf3a9a58e75bf06d2cf7d8fbcd5"},"schema_version":"1.0","source":{"id":"1501.00573","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00573","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00573v1","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00573","created_at":"2026-05-18T02:30:07Z"},{"alias_kind":"pith_short_12","alias_value":"UCGDQJK7MK7A","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UCGDQJK7MK7A3BR2","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UCGDQJK7","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:cc0b897cc7f2c3a126cb96bbd10a400a858a76e8f0285daae82025c44f3c8283","target":"graph","created_at":"2026-05-18T02:30:07Z","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":"The numbers $R_n$ and $W_n$ are defined as \\begin{align*} R_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{1}{2k-1},\\ \\text{and}\\ W_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{3}{2k-3}. \\end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \\begin{align*} \\sum_{k=0}^{n-1}(2k+1)R_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)R_k^2 &\\equiv 4p(-1)^{\\frac{p-1}{2}} -p^2 \\pmod{p^3}, \\\\ 9\\sum_{k=0}^{n-1}(2k+1)W_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)W_k^2 &\\equiv 12p(-1)^{\\frac{p-1}{2}}-17p^2 \\pmod{p^3}, \\quad\\text{if $p>3$.} \\end{align*} The firs","authors_text":"Ji-Cai Liu, Victor J. W. Guo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","title":"Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00573","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":"sha256:339c89c033435e670bb5d23e0b4a774b033fad7936ebd7168b7737bf88a5fac8","target":"record","created_at":"2026-05-18T02:30:07Z","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":"2bdcda7acc289a88853bd0e394b2362f019d48fc0a8daf86484c22c1f524be65","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","title_canon_sha256":"b2cdea41e556fcfb9ba956745e12e5d721bf7bf3a9a58e75bf06d2cf7d8fbcd5"},"schema_version":"1.0","source":{"id":"1501.00573","kind":"arxiv","version":1}},"canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","first_computed_at":"2026-05-18T02:30:07.211045Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:07.211045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bc0kOelAidIaH3r+c31/B2IWosHp3AWrXE9VUb5rgpxPXZATAQOENoQY4fUtKSge76EjnO9C0ehoYHzYenLaAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:07.211780Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.00573","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:339c89c033435e670bb5d23e0b4a774b033fad7936ebd7168b7737bf88a5fac8","sha256:cc0b897cc7f2c3a126cb96bbd10a400a858a76e8f0285daae82025c44f3c8283"],"state_sha256":"19395d5e9d01d72561826d8b69176b9317efc3f267a4171074685237ee2c79e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oAUZfQBOj0ecA/CgKnHeqykcBZOj+1+ieztIyA3VhiDkQQUwyKdhhXtAR+1wfk+YPDJ8CVd4O+fIz8exQ7Z9BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T02:47:03.322570Z","bundle_sha256":"3c7663b7ba01f11188a6a17b0b6eeb787a2a98297df1e973b0ac276059b58f3c"}}