{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:V4B7N3FKXA3CHQ3Q4HI6TOKQJZ","short_pith_number":"pith:V4B7N3FK","canonical_record":{"source":{"id":"1604.05019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-18T07:22:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"966297c5a5e6ef4958338cc66e9bdaafd2ca23c978f00d12163fff7992a160db","abstract_canon_sha256":"9f6877108bcd28cc4e450797e2266b5015874881384c543294334375178070f1"},"schema_version":"1.0"},"canonical_sha256":"af03f6ecaab83623c370e1d1e9b9504e42da73b3c047b81689259a8bf28d8636","source":{"kind":"arxiv","id":"1604.05019","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05019","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05019v1","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05019","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"pith_short_12","alias_value":"V4B7N3FKXA3C","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4B7N3FKXA3CHQ3Q","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4B7N3FK","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:V4B7N3FKXA3CHQ3Q4HI6TOKQJZ","target":"record","payload":{"canonical_record":{"source":{"id":"1604.05019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-18T07:22:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"966297c5a5e6ef4958338cc66e9bdaafd2ca23c978f00d12163fff7992a160db","abstract_canon_sha256":"9f6877108bcd28cc4e450797e2266b5015874881384c543294334375178070f1"},"schema_version":"1.0"},"canonical_sha256":"af03f6ecaab83623c370e1d1e9b9504e42da73b3c047b81689259a8bf28d8636","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:56.035157Z","signature_b64":"BKlKib653T6KVv5AGm4LxidldoI/6EcUrdvTrS5JefRMWCQBsTbUiUC33T5hb6lfP//ZDK8EMWxQwq1GVdRnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af03f6ecaab83623c370e1d1e9b9504e42da73b3c047b81689259a8bf28d8636","last_reissued_at":"2026-05-18T01:16:56.034622Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:56.034622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.05019","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":"2026-05-18T01:16:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l8V6allf1jzi1cUY73aPeQdr8Y8ioZOPMPLR70CLIrza50eVXDxPGFRKpOs86H2A0evK7JEAnaJYNJxQMfhoBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T04:43:36.473708Z"},"content_sha256":"7286da2842e2c7c0ac24694081a9b7fa6c86b9375b4b7812d40583a732bbfe35","schema_version":"1.0","event_id":"sha256:7286da2842e2c7c0ac24694081a9b7fa6c86b9375b4b7812d40583a732bbfe35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:V4B7N3FKXA3CHQ3Q4HI6TOKQJZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of a recent conjecture of Z.-W. Sun","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Song Guo, Victor J. W. Guo","submitted_at":"2016-04-18T07:22:38Z","abstract_excerpt":"The polynomials $d_n(x)$ are defined by \\begin{align*} d_n(x) &= \\sum_{k=0}^n{n\\choose k}{x\\choose k}2^k. \\end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \\begin{align*} \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{4}\\right)^2 &\\equiv \\begin{cases} 2(-1)^{\\frac{p-1}{4}}x,&\\text{if $p=x^2+y^2$ with $x\\equiv 1\\pmod{4}$,} 0,&\\text{if $p\\equiv 3\\pmod{4}$,} \\end{cases} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{6}\\right)^2 &\\equiv 0, \\quad\\text{if $p>3$,} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(\\frac{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05019","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":"2026-05-18T01:16:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OBFJV7cxNyUwrBsMrXJOZ97n3+QxyUyoHQGei4oELl3mzAWRdvYi0v8yvzngPkJyG+WbvOTh/gYC/4bbMlI1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T04:43:36.474460Z"},"content_sha256":"25df31702261c460f8a36d7bd6ae401874d8e78cbac1dd8d7136e624af626a42","schema_version":"1.0","event_id":"sha256:25df31702261c460f8a36d7bd6ae401874d8e78cbac1dd8d7136e624af626a42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/bundle.json","state_url":"https://pith.science/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/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":"2026-05-26T04:43:36Z","links":{"resolver":"https://pith.science/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ","bundle":"https://pith.science/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/bundle.json","state":"https://pith.science/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4B7N3FKXA3CHQ3Q4HI6TOKQJZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:V4B7N3FKXA3CHQ3Q4HI6TOKQJZ","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":"9f6877108bcd28cc4e450797e2266b5015874881384c543294334375178070f1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-18T07:22:38Z","title_canon_sha256":"966297c5a5e6ef4958338cc66e9bdaafd2ca23c978f00d12163fff7992a160db"},"schema_version":"1.0","source":{"id":"1604.05019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05019","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05019v1","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05019","created_at":"2026-05-18T01:16:56Z"},{"alias_kind":"pith_short_12","alias_value":"V4B7N3FKXA3C","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4B7N3FKXA3CHQ3Q","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4B7N3FK","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:25df31702261c460f8a36d7bd6ae401874d8e78cbac1dd8d7136e624af626a42","target":"graph","created_at":"2026-05-18T01:16:56Z","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 polynomials $d_n(x)$ are defined by \\begin{align*} d_n(x) &= \\sum_{k=0}^n{n\\choose k}{x\\choose k}2^k. \\end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \\begin{align*} \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{4}\\right)^2 &\\equiv \\begin{cases} 2(-1)^{\\frac{p-1}{4}}x,&\\text{if $p=x^2+y^2$ with $x\\equiv 1\\pmod{4}$,} 0,&\\text{if $p\\equiv 3\\pmod{4}$,} \\end{cases} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(-\\frac{1}{6}\\right)^2 &\\equiv 0, \\quad\\text{if $p>3$,} [5pt] \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}}{4^k} d_k\\left(\\frac{1","authors_text":"Song Guo, Victor J. W. Guo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-18T07:22:38Z","title":"Proof of a recent conjecture of Z.-W. Sun"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05019","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":"sha256:7286da2842e2c7c0ac24694081a9b7fa6c86b9375b4b7812d40583a732bbfe35","target":"record","created_at":"2026-05-18T01:16:56Z","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":"9f6877108bcd28cc4e450797e2266b5015874881384c543294334375178070f1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-18T07:22:38Z","title_canon_sha256":"966297c5a5e6ef4958338cc66e9bdaafd2ca23c978f00d12163fff7992a160db"},"schema_version":"1.0","source":{"id":"1604.05019","kind":"arxiv","version":1}},"canonical_sha256":"af03f6ecaab83623c370e1d1e9b9504e42da73b3c047b81689259a8bf28d8636","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af03f6ecaab83623c370e1d1e9b9504e42da73b3c047b81689259a8bf28d8636","first_computed_at":"2026-05-18T01:16:56.034622Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:56.034622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BKlKib653T6KVv5AGm4LxidldoI/6EcUrdvTrS5JefRMWCQBsTbUiUC33T5hb6lfP//ZDK8EMWxQwq1GVdRnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:56.035157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.05019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7286da2842e2c7c0ac24694081a9b7fa6c86b9375b4b7812d40583a732bbfe35","sha256:25df31702261c460f8a36d7bd6ae401874d8e78cbac1dd8d7136e624af626a42"],"state_sha256":"fd296ecf29c1416685f67fda6018c79f761147ea9440f82dfdd11bb05ba084a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m3A4mOo8YxdkLvPVjKA7QYe7b1NuwjvekhSjVTUlRHXxYrnE8rxQM9zCGFyaXolxXZhPy7FJb164l020y8NOCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T04:43:36.479184Z","bundle_sha256":"7ffc3e438fb3788306cb1f3ab380b6b8048ca719c55229accd5ff393a4129105"}}