{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:N54FJSH45T6GCLY7Q7MRM6JK5A","short_pith_number":"pith:N54FJSH4","canonical_record":{"source":{"id":"2408.14401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-08-26T16:34:29Z","cross_cats_sorted":[],"title_canon_sha256":"818dd56697d9c6f63909ca69b405cf9d341f9b640506dbdbf409d1c4d2078419","abstract_canon_sha256":"612de79b3e0f2d28a49835dcd87d82bbeb5bc7d877b6fa7e6e6097884eeab7f0"},"schema_version":"1.0"},"canonical_sha256":"6f7854c8fcecfc612f1f87d916792ae82143d2afff5c67d7160c69b8564c9021","source":{"kind":"arxiv","id":"2408.14401","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.14401","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"2408.14401v1","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.14401","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"N54FJSH45T6G","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"N54FJSH45T6GCLY7","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"N54FJSH4","created_at":"2026-07-05T08:59:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:N54FJSH45T6GCLY7Q7MRM6JK5A","target":"record","payload":{"canonical_record":{"source":{"id":"2408.14401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-08-26T16:34:29Z","cross_cats_sorted":[],"title_canon_sha256":"818dd56697d9c6f63909ca69b405cf9d341f9b640506dbdbf409d1c4d2078419","abstract_canon_sha256":"612de79b3e0f2d28a49835dcd87d82bbeb5bc7d877b6fa7e6e6097884eeab7f0"},"schema_version":"1.0"},"canonical_sha256":"6f7854c8fcecfc612f1f87d916792ae82143d2afff5c67d7160c69b8564c9021","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:59:20.241660Z","signature_b64":"uyJ6jcndpL0rfW7SYsGb5KoEbpNerk+ZlQjBOQDW7D5j7Whldv7lTH6eVLfp+jPwnTNkuSoc7ivu2rWUnWb6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f7854c8fcecfc612f1f87d916792ae82143d2afff5c67d7160c69b8564c9021","last_reissued_at":"2026-07-05T08:59:20.241182Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:59:20.241182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2408.14401","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":"2026-07-05T08:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f9OJdMwWYbc6YhSYMxDxcevYB4f/qrM0xd/eohI7af5JfOLTfQaOHw0hySt6rpIbyEYpfsiuiLCJRONQvMmHCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T15:55:42.528420Z"},"content_sha256":"ced3266f79f5740e380d327452822ad08f744fb805c190794f3ed4bbada89dcc","schema_version":"1.0","event_id":"sha256:ced3266f79f5740e380d327452822ad08f744fb805c190794f3ed4bbada89dcc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:N54FJSH45T6GCLY7Q7MRM6JK5A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On determinants involving $(\\frac{j^2-k^2}p)$ and $(\\frac{jk}p)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Deyi Chen, Zhi-Wei Sun","submitted_at":"2024-08-26T16:34:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $(\\frac{\\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\\det\\left[\\left(\\frac{j^2-k^2}p\\right)+\\left(\\frac{jk}p\\right)w\\right]_{\\delta\\le j,k\\le (p-1)/2}$$ with $\\delta\\in\\{0,1\\}$. For example, we prove that the determinant does not depend on $w$ if $p\\equiv3\\pmod4$ and $\\delta=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.14401","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/2408.14401/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":"2026-07-05T08:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qtgU6mQimXmrFADD2F4GcPpKBMj+Zz7xGjfgLVSTQQ+Lc6/wyLFzd9zIoCpVc3I0UB/DAYiTBlnNfj+KHN3AAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T15:55:42.530273Z"},"content_sha256":"034899aa6596f05245d21022f089f5c1c1638e07dfd0cf98396c1d5719058946","schema_version":"1.0","event_id":"sha256:034899aa6596f05245d21022f089f5c1c1638e07dfd0cf98396c1d5719058946"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/bundle.json","state_url":"https://pith.science/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/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":"2026-08-07T15:55:42Z","links":{"resolver":"https://pith.science/pith/N54FJSH45T6GCLY7Q7MRM6JK5A","bundle":"https://pith.science/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/bundle.json","state":"https://pith.science/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N54FJSH45T6GCLY7Q7MRM6JK5A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:N54FJSH45T6GCLY7Q7MRM6JK5A","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":"612de79b3e0f2d28a49835dcd87d82bbeb5bc7d877b6fa7e6e6097884eeab7f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-08-26T16:34:29Z","title_canon_sha256":"818dd56697d9c6f63909ca69b405cf9d341f9b640506dbdbf409d1c4d2078419"},"schema_version":"1.0","source":{"id":"2408.14401","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.14401","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"2408.14401v1","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.14401","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"N54FJSH45T6G","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"N54FJSH45T6GCLY7","created_at":"2026-07-05T08:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"N54FJSH4","created_at":"2026-07-05T08:59:20Z"}],"graph_snapshots":[{"event_id":"sha256:034899aa6596f05245d21022f089f5c1c1638e07dfd0cf98396c1d5719058946","target":"graph","created_at":"2026-07-05T08:59:20Z","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/2408.14401/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p$ be an odd prime and let $(\\frac{\\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\\det\\left[\\left(\\frac{j^2-k^2}p\\right)+\\left(\\frac{jk}p\\right)w\\right]_{\\delta\\le j,k\\le (p-1)/2}$$ with $\\delta\\in\\{0,1\\}$. For example, we prove that the determinant does not depend on $w$ if $p\\equiv3\\pmod4$ and $\\delta=0$.","authors_text":"Deyi Chen, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-08-26T16:34:29Z","title":"On determinants involving $(\\frac{j^2-k^2}p)$ and $(\\frac{jk}p)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.14401","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:ced3266f79f5740e380d327452822ad08f744fb805c190794f3ed4bbada89dcc","target":"record","created_at":"2026-07-05T08:59:20Z","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":"612de79b3e0f2d28a49835dcd87d82bbeb5bc7d877b6fa7e6e6097884eeab7f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-08-26T16:34:29Z","title_canon_sha256":"818dd56697d9c6f63909ca69b405cf9d341f9b640506dbdbf409d1c4d2078419"},"schema_version":"1.0","source":{"id":"2408.14401","kind":"arxiv","version":1}},"canonical_sha256":"6f7854c8fcecfc612f1f87d916792ae82143d2afff5c67d7160c69b8564c9021","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f7854c8fcecfc612f1f87d916792ae82143d2afff5c67d7160c69b8564c9021","first_computed_at":"2026-07-05T08:59:20.241182Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:59:20.241182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uyJ6jcndpL0rfW7SYsGb5KoEbpNerk+ZlQjBOQDW7D5j7Whldv7lTH6eVLfp+jPwnTNkuSoc7ivu2rWUnWb6Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:59:20.241660Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.14401","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ced3266f79f5740e380d327452822ad08f744fb805c190794f3ed4bbada89dcc","sha256:034899aa6596f05245d21022f089f5c1c1638e07dfd0cf98396c1d5719058946"],"state_sha256":"61dc96aef30e776a6bc004edde29ce2ff806a39c9d8217f8c59b462b664fe014"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F5tLIwVG0JDPR8lM7lN/uDgjh9K+CZjb1pAPiHeC/4txrSXwj54a0w6+zW99gBfowQBQu217JXG0Uh5dLb2mAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T15:55:42.538135Z","bundle_sha256":"932ee3eabb4c1fa9303a97f7ec55e9a07f2c042077155b84014dd2e3fddb4446"}}