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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."},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2404.11547","created_at":"2026-07-05T08:09:10.317267+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.11547v1","created_at":"2026-07-05T08:09:10.317267+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.11547","created_at":"2026-07-05T08:09:10.317267+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z4KR6KU53CSL","created_at":"2026-07-05T08:09:10.317267+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z4KR6KU53CSL77AP","created_at":"2026-07-05T08:09:10.317267+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z4KR6KU5","created_at":"2026-07-05T08:09:10.317267+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/Z4KR6KU53CSL77APKDIUFHJJ65","json":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65.json","graph_json":"https://pith.science/api/pith-number/Z4KR6KU53CSL77APKDIUFHJJ65/graph.json","events_json":"https://pith.science/api/pith-number/Z4KR6KU53CSL77APKDIUFHJJ65/events.json","paper":"https://pith.science/paper/Z4KR6KU5"},"agent_actions":{"view_html":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65","download_json":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65.json","view_paper":"https://pith.science/paper/Z4KR6KU5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.11547&json=true","fetch_graph":"https://pith.science/api/pith-number/Z4KR6KU53CSL77APKDIUFHJJ65/graph.json","fetch_events":"https://pith.science/api/pith-number/Z4KR6KU53CSL77APKDIUFHJJ65/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/action/storage_attestation","attest_author":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/action/author_attestation","sign_citation":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/action/citation_signature","submit_replication":"https://pith.science/pith/Z4KR6KU53CSL77APKDIUFHJJ65/action/replication_record"}},"created_at":"2026-07-05T08:09:10.317267+00:00","updated_at":"2026-07-05T08:09:10.317267+00:00"}