{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:HED7SE63IZMP32AFXJQ6ZDDPIP","short_pith_number":"pith:HED7SE63","schema_version":"1.0","canonical_sha256":"3907f913db4658fde805ba61ec8c6f43ce0ef214f51eeb3dac2a70b10b041e16","source":{"kind":"arxiv","id":"1812.08080","version":5},"attestation_state":"computed","paper":{"title":"On some determinants involving Jacobi symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitry Krachun, Fedor Petrov, Maxim Vsemirnov, Zhi-Wei Sun","submitted_at":"2018-12-19T16:54:15Z","abstract_excerpt":"In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\\equiv3\\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\\bigg|\\left(\\frac{i^2+cij+dj^2}n\\right)\\bigg|_{1\\le i,j\\le n-1}$$ and $$[c,d]_n=\\bigg|\\left(\\frac{i^2+cij+dj^2}n\\right)\\bigg|_{0\\le i,j\\le n-1}$$ with $(\\frac{\\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\\equiv5\\pmod{12}$, and $[5,5]_p=0$ for any prime $p\\equiv 13,17\\pmod{20}$, which we"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.08080","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-19T16:54:15Z","cross_cats_sorted":[],"title_canon_sha256":"157ef297b015be3d86bdddb11a73bf752a2a7493d93daa8ebdb2071d6a7982cd","abstract_canon_sha256":"8998fed5428d9b6e9a4bd0cefd83d11878cf25a505c792afec79fbbf20c66379"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:49:35.216405Z","signature_b64":"pDHLpxymi0+Xjb0/5HLixUyQB90T39UpGLcyR+nQ9QA/IA8L9zypgF7HTvIVUvU0E+NOWY3jIvgvPszjxRlwAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3907f913db4658fde805ba61ec8c6f43ce0ef214f51eeb3dac2a70b10b041e16","last_reissued_at":"2026-07-05T00:49:35.215828Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:49:35.215828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some determinants involving Jacobi symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitry Krachun, Fedor Petrov, Maxim Vsemirnov, Zhi-Wei Sun","submitted_at":"2018-12-19T16:54:15Z","abstract_excerpt":"In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\\equiv3\\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\\bigg|\\left(\\frac{i^2+cij+dj^2}n\\right)\\bigg|_{1\\le i,j\\le n-1}$$ and $$[c,d]_n=\\bigg|\\left(\\frac{i^2+cij+dj^2}n\\right)\\bigg|_{0\\le i,j\\le n-1}$$ with $(\\frac{\\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\\equiv5\\pmod{12}$, and $[5,5]_p=0$ for any prime $p\\equiv 13,17\\pmod{20}$, which we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08080","kind":"arxiv","version":5},"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/1812.08080/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1812.08080","created_at":"2026-07-05T00:49:35.215887+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.08080v5","created_at":"2026-07-05T00:49:35.215887+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08080","created_at":"2026-07-05T00:49:35.215887+00:00"},{"alias_kind":"pith_short_12","alias_value":"HED7SE63IZMP","created_at":"2026-07-05T00:49:35.215887+00:00"},{"alias_kind":"pith_short_16","alias_value":"HED7SE63IZMP32AF","created_at":"2026-07-05T00:49:35.215887+00:00"},{"alias_kind":"pith_short_8","alias_value":"HED7SE63","created_at":"2026-07-05T00:49:35.215887+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP","json":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP.json","graph_json":"https://pith.science/api/pith-number/HED7SE63IZMP32AFXJQ6ZDDPIP/graph.json","events_json":"https://pith.science/api/pith-number/HED7SE63IZMP32AFXJQ6ZDDPIP/events.json","paper":"https://pith.science/paper/HED7SE63"},"agent_actions":{"view_html":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP","download_json":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP.json","view_paper":"https://pith.science/paper/HED7SE63","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.08080&json=true","fetch_graph":"https://pith.science/api/pith-number/HED7SE63IZMP32AFXJQ6ZDDPIP/graph.json","fetch_events":"https://pith.science/api/pith-number/HED7SE63IZMP32AFXJQ6ZDDPIP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP/action/storage_attestation","attest_author":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP/action/author_attestation","sign_citation":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP/action/citation_signature","submit_replication":"https://pith.science/pith/HED7SE63IZMP32AFXJQ6ZDDPIP/action/replication_record"}},"created_at":"2026-07-05T00:49:35.215887+00:00","updated_at":"2026-07-05T00:49:35.215887+00:00"}