{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YKLRQPZXSPFX2MOWRRPJRHM3Y2","short_pith_number":"pith:YKLRQPZX","schema_version":"1.0","canonical_sha256":"c297183f3793cb7d31d68c5e989d9bc6b28742dd8f1739302a6898f2ab89dca9","source":{"kind":"arxiv","id":"2605.19486","version":1},"attestation_state":"computed","paper":{"title":"A Determinant Congruence Conjectured by Sun","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-19T07:37:46Z","abstract_excerpt":"We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\\in\\Z$. If $n$ is composite, then \\[\n  \\det\\big[(i^2+cij+dj^2)^{n-2}\\big]_{0\\leq i,j\\leq n-1}\\equiv 0\\pmod {n^2} \\] with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\\leg{d}{p}$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p"},"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":"2605.19486","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-19T07:37:46Z","cross_cats_sorted":[],"title_canon_sha256":"bc39ee45aa657f914cfd10ed3131e54de3388992fe9491442711266e2b2e5011","abstract_canon_sha256":"3b3d9e479eaae7f20af94b4fc445d86ab7d2a91a0058b5e14e11ce6c5a882b12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:48.123773Z","signature_b64":"h7kcjSkkxYZvrDPhdZ1Kgt/QjkYKx5UU74iMTK691eg3L/AXsx3RYQ1dTN1YEcK+zuX4KeI4emfW9SnOPmL5Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c297183f3793cb7d31d68c5e989d9bc6b28742dd8f1739302a6898f2ab89dca9","last_reissued_at":"2026-05-20T01:05:48.123060Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:48.123060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Determinant Congruence Conjectured by Sun","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-19T07:37:46Z","abstract_excerpt":"We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\\in\\Z$. If $n$ is composite, then \\[\n  \\det\\big[(i^2+cij+dj^2)^{n-2}\\big]_{0\\leq i,j\\leq n-1}\\equiv 0\\pmod {n^2} \\] with no condition on $c$ and $d$. If $n=p$ is prime, the same congruence holds whenever the Legendre symbol $\\leg{d}{p}$ is $-1$. For composite $n$, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime $n=p$, we estimate the rank of the matrix modulo $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19486","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/2605.19486/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":"2605.19486","created_at":"2026-05-20T01:05:48.123190+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.19486v1","created_at":"2026-05-20T01:05:48.123190+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19486","created_at":"2026-05-20T01:05:48.123190+00:00"},{"alias_kind":"pith_short_12","alias_value":"YKLRQPZXSPFX","created_at":"2026-05-20T01:05:48.123190+00:00"},{"alias_kind":"pith_short_16","alias_value":"YKLRQPZXSPFX2MOW","created_at":"2026-05-20T01:05:48.123190+00:00"},{"alias_kind":"pith_short_8","alias_value":"YKLRQPZX","created_at":"2026-05-20T01:05:48.123190+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/YKLRQPZXSPFX2MOWRRPJRHM3Y2","json":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2.json","graph_json":"https://pith.science/api/pith-number/YKLRQPZXSPFX2MOWRRPJRHM3Y2/graph.json","events_json":"https://pith.science/api/pith-number/YKLRQPZXSPFX2MOWRRPJRHM3Y2/events.json","paper":"https://pith.science/paper/YKLRQPZX"},"agent_actions":{"view_html":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2","download_json":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2.json","view_paper":"https://pith.science/paper/YKLRQPZX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.19486&json=true","fetch_graph":"https://pith.science/api/pith-number/YKLRQPZXSPFX2MOWRRPJRHM3Y2/graph.json","fetch_events":"https://pith.science/api/pith-number/YKLRQPZXSPFX2MOWRRPJRHM3Y2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2/action/storage_attestation","attest_author":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2/action/author_attestation","sign_citation":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2/action/citation_signature","submit_replication":"https://pith.science/pith/YKLRQPZXSPFX2MOWRRPJRHM3Y2/action/replication_record"}},"created_at":"2026-05-20T01:05:48.123190+00:00","updated_at":"2026-05-20T01:05:48.123190+00:00"}