{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:Z4KR6KU53CSL77APKDIUFHJJ65","short_pith_number":"pith:Z4KR6KU5","canonical_record":{"source":{"id":"2404.11547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","cross_cats_sorted":[],"title_canon_sha256":"4230feb43512ec1d621a637b29c1b8ce7a8edc2b7aa8502095e8a1b61c1e3f4d","abstract_canon_sha256":"16c541a50b4841a43c2ec1ecc66ad06c312cb829d92fec47e0179ee3fac5f533"},"schema_version":"1.0"},"canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","source":{"kind":"arxiv","id":"2404.11547","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.11547","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"arxiv_version","alias_value":"2404.11547v1","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.11547","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_12","alias_value":"Z4KR6KU53CSL","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_16","alias_value":"Z4KR6KU53CSL77AP","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_8","alias_value":"Z4KR6KU5","created_at":"2026-07-05T08:09:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:Z4KR6KU53CSL77APKDIUFHJJ65","target":"record","payload":{"canonical_record":{"source":{"id":"2404.11547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","cross_cats_sorted":[],"title_canon_sha256":"4230feb43512ec1d621a637b29c1b8ce7a8edc2b7aa8502095e8a1b61c1e3f4d","abstract_canon_sha256":"16c541a50b4841a43c2ec1ecc66ad06c312cb829d92fec47e0179ee3fac5f533"},"schema_version":"1.0"},"canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:09:10.317732Z","signature_b64":"BNZfU5/+w5WLEYx+nLvBfd8YfUDy85bfmdCD+HFpe+K3PxhrG7/KXuOCxOXRLv9sNiM4BnHfHqcNU9bDov+qDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","last_reissued_at":"2026-07-05T08:09:10.317193Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:09:10.317193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.11547","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":"2026-07-05T08:09:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3/XNuZh+NbA+YUyrkZYDSbZbQXx0Q+uHEtykl3vCCAwXlZ7APf4cBpAL0xEEvgS8GYy0NrhYivj+tyyYuS3LAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T02:21:46.971344Z"},"content_sha256":"a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5","schema_version":"1.0","event_id":"sha256:a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:Z4KR6KU53CSL77APKDIUFHJJ65","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On some determinants arising from quadratic residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chen-Kai Ren, Zhi-Wei Sun","submitted_at":"2024-04-17T16:43:17Z","abstract_excerpt":"Let $p>3$ be a prime, and let $d\\in\\mathbb Z$ with $p\\nmid d$. For the determinants $$S_m(d,p)=\\det\\left[(i^2+dj^2)^{m}\\right]_{1\\leqslant i,j \\leqslant (p-1)/2}\\ \\ \\left(\\frac{p-1}2\\leqslant m\\leqslant p-1\\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\\in\\{p-2,p-3\\}$ and $(\\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\\frac{-d}p)=1$, and determine the Legendre symbols $(\\frac{S_{p-3}\\,(d,p)}p)$ and $(\\frac{S_{p-4}\\,(d,p)}p)$ in some special cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.11547","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.11547/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:09:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"arHNFb4MmK3hYBSFjKXdrP6729nCwG5xFQH3XYj7fKKbBFsFokZivJLrGmOLkyCs9bGEIGfmn+slWoqUyvGxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T02:21:46.971930Z"},"content_sha256":"d7cd1553e7a0e781833dae1d6211213b8ede01c35a2189ef7af386739dd73de2","schema_version":"1.0","event_id":"sha256:d7cd1553e7a0e781833dae1d6211213b8ede01c35a2189ef7af386739dd73de2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/bundle.json","state_url":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z4KR6KU53CSL77APKDIUFHJJ65/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":"2026-08-04T02:21:46Z","links":{"resolver":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65","bundle":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/bundle.json","state":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z4KR6KU53CSL77APKDIUFHJJ65/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:Z4KR6KU53CSL77APKDIUFHJJ65","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":"16c541a50b4841a43c2ec1ecc66ad06c312cb829d92fec47e0179ee3fac5f533","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","title_canon_sha256":"4230feb43512ec1d621a637b29c1b8ce7a8edc2b7aa8502095e8a1b61c1e3f4d"},"schema_version":"1.0","source":{"id":"2404.11547","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.11547","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"arxiv_version","alias_value":"2404.11547v1","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.11547","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_12","alias_value":"Z4KR6KU53CSL","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_16","alias_value":"Z4KR6KU53CSL77AP","created_at":"2026-07-05T08:09:10Z"},{"alias_kind":"pith_short_8","alias_value":"Z4KR6KU5","created_at":"2026-07-05T08:09:10Z"}],"graph_snapshots":[{"event_id":"sha256:d7cd1553e7a0e781833dae1d6211213b8ede01c35a2189ef7af386739dd73de2","target":"graph","created_at":"2026-07-05T08:09:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2404.11547/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p>3$ be a prime, and let $d\\in\\mathbb Z$ with $p\\nmid d$. For the determinants $$S_m(d,p)=\\det\\left[(i^2+dj^2)^{m}\\right]_{1\\leqslant i,j \\leqslant (p-1)/2}\\ \\ \\left(\\frac{p-1}2\\leqslant m\\leqslant p-1\\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\\in\\{p-2,p-3\\}$ and $(\\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\\frac{-d}p)=1$, and determine the Legendre symbols $(\\frac{S_{p-3}\\,(d,p)}p)$ and $(\\frac{S_{p-4}\\,(d,p)}p)$ in some special cases.","authors_text":"Chen-Kai Ren, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","title":"On some determinants arising from quadratic residues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.11547","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:a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5","target":"record","created_at":"2026-07-05T08:09:10Z","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":"16c541a50b4841a43c2ec1ecc66ad06c312cb829d92fec47e0179ee3fac5f533","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","title_canon_sha256":"4230feb43512ec1d621a637b29c1b8ce7a8edc2b7aa8502095e8a1b61c1e3f4d"},"schema_version":"1.0","source":{"id":"2404.11547","kind":"arxiv","version":1}},"canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","first_computed_at":"2026-07-05T08:09:10.317193Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:09:10.317193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BNZfU5/+w5WLEYx+nLvBfd8YfUDy85bfmdCD+HFpe+K3PxhrG7/KXuOCxOXRLv9sNiM4BnHfHqcNU9bDov+qDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:09:10.317732Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.11547","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5","sha256:d7cd1553e7a0e781833dae1d6211213b8ede01c35a2189ef7af386739dd73de2"],"state_sha256":"b8f29642d294aa60344b1532e776aa715cd11f1cec8148020b9285c007a3a248"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8MG4SoQkHx6c4SxwSFFfytLAsZB53UpZlrergTN/egvgVIzrXKvr5NJFpIWyom4tpoO1v9UxGU4bBd8DXx9CBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T02:21:46.977132Z","bundle_sha256":"e023ff226371e4803dfb60222941f89a80ec974f45890a2d052534dce633b83f"}}