{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:IW3JVZWWTBR5FG2ASI6EGSUNCI","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":"479cdfa3fd8a1caa7f0447ea43a735bda5b5a9bcedf90a755c33489f4e09bb16","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-20T15:14:34Z","title_canon_sha256":"b6502027ee940cddc6e187b75f34d7e4b1450cd10a75b192fbacdd8e428bfe06"},"schema_version":"1.0","source":{"id":"1809.07766","kind":"arxiv","version":10}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07766","created_at":"2026-05-17T23:41:12Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07766v10","created_at":"2026-05-17T23:41:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07766","created_at":"2026-05-17T23:41:12Z"},{"alias_kind":"pith_short_12","alias_value":"IW3JVZWWTBR5","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"IW3JVZWWTBR5FG2A","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"IW3JVZWW","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:b8fbd1a8f8b4b918abd63679ed97375905efc23bcb0958bc5de5cd1007ff239b","target":"graph","created_at":"2026-05-17T23:41:12Z","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":"Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\\ldots,p-1$, then the list $\\{1^2\\}_p,\\ldots,\\{((p-1)/2)^2\\}_p$ (with $\\{k\\}_p$ the least nonnegative residue of $k$ modulo $p$) is a permutation of $a_1,\\ldots,a_{(p-1)/2}$, and we show that the sign of this permutation is $1$ or $(-1)^{(h(-p)+1)/2}$ according as $p\\equiv3\\pmod 8$ or $p\\equiv7\\pmod 8$, where $h(-p)$ is the class number of the imaginary quadratic field $\\mathbb Q(","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":"2018-09-20T15:14:34Z","title":"Quadratic residues and related permutations and identities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07766","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":"sha256:5b1389c241892be6c68936b05b651a30790d07f3ea7995361940d777c45be69b","target":"record","created_at":"2026-05-17T23:41:12Z","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":"479cdfa3fd8a1caa7f0447ea43a735bda5b5a9bcedf90a755c33489f4e09bb16","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-20T15:14:34Z","title_canon_sha256":"b6502027ee940cddc6e187b75f34d7e4b1450cd10a75b192fbacdd8e428bfe06"},"schema_version":"1.0","source":{"id":"1809.07766","kind":"arxiv","version":10}},"canonical_sha256":"45b69ae6d69863d29b40923c434a8d120d83e33b4c2747c2005e62b1bd60d915","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45b69ae6d69863d29b40923c434a8d120d83e33b4c2747c2005e62b1bd60d915","first_computed_at":"2026-05-17T23:41:12.917892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:12.917892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G8Oyzf5c6076LSWNcdbHxE2ZnZVMHvforRFT6ksXZEQOyH2147DgRYj65tzjlkzp9WCCr9heQJA2ifQdaZeJBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:12.918371Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.07766","source_kind":"arxiv","source_version":10}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b1389c241892be6c68936b05b651a30790d07f3ea7995361940d777c45be69b","sha256:b8fbd1a8f8b4b918abd63679ed97375905efc23bcb0958bc5de5cd1007ff239b"],"state_sha256":"c1601601f08ef9fe64c7bef2225dcf69f2eae6da8018a6e1fc30b785cb435d2b"}