{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2003:MOFYXRZ76XTOK3PKSNPJA2B6F7","short_pith_number":"pith:MOFYXRZ7","schema_version":"1.0","canonical_sha256":"638b8bc73ff5e6e56dea935e90683e2fda202bf23eb609032a7a9c8887e249e1","source":{"kind":"arxiv","id":"math/0312010","version":3},"attestation_state":"computed","paper":{"title":"Simple arguments on consecutive power residues","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2003-11-29T13:42:51Z","abstract_excerpt":"By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ring of algebraic integers in a quadratic field $K=Q(\\sqrt d)$ with d in {-1,-2,-3,-7,-11}. Then, for any irreducible $\\pi\\in O_K$ and positive integer k not relatively prime to $\\pi\\bar\\pi-1$, there exists a k-th power non-residue $\\omega\\in O_K$ modulo $\\pi$ such that $|\\omega|<\\sqrt{|\\pi|}+0.65$."},"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":"math/0312010","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2003-11-29T13:42:51Z","cross_cats_sorted":[],"title_canon_sha256":"cbc282de0631cfeb7afd3ff8cf2dcf547a7f01e0fe99f86fbdd1243480c439bb","abstract_canon_sha256":"c166e58ab8fe226ad59be62bbb4eb95d465ff3f83f28f2d0196d05cf90a17339"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:59:38.910758Z","signature_b64":"tGdwww63xfHZ+ZuvAZfkTeK8NX0FbLyzriDsvNI/S0SSN+0MWxmTyQ5dleKCrtObmZAnfebVTC3Fz1q0UUJFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"638b8bc73ff5e6e56dea935e90683e2fda202bf23eb609032a7a9c8887e249e1","last_reissued_at":"2026-07-04T14:59:38.910376Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:59:38.910376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simple arguments on consecutive power residues","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2003-11-29T13:42:51Z","abstract_excerpt":"By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ring of algebraic integers in a quadratic field $K=Q(\\sqrt d)$ with d in {-1,-2,-3,-7,-11}. Then, for any irreducible $\\pi\\in O_K$ and positive integer k not relatively prime to $\\pi\\bar\\pi-1$, there exists a k-th power non-residue $\\omega\\in O_K$ modulo $\\pi$ such that $|\\omega|<\\sqrt{|\\pi|}+0.65$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0312010","kind":"arxiv","version":3},"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/math/0312010/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":"math/0312010","created_at":"2026-07-04T14:59:38.910440+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0312010v3","created_at":"2026-07-04T14:59:38.910440+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0312010","created_at":"2026-07-04T14:59:38.910440+00:00"},{"alias_kind":"pith_short_12","alias_value":"MOFYXRZ76XTO","created_at":"2026-07-04T14:59:38.910440+00:00"},{"alias_kind":"pith_short_16","alias_value":"MOFYXRZ76XTOK3PK","created_at":"2026-07-04T14:59:38.910440+00:00"},{"alias_kind":"pith_short_8","alias_value":"MOFYXRZ7","created_at":"2026-07-04T14:59:38.910440+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/MOFYXRZ76XTOK3PKSNPJA2B6F7","json":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7.json","graph_json":"https://pith.science/api/pith-number/MOFYXRZ76XTOK3PKSNPJA2B6F7/graph.json","events_json":"https://pith.science/api/pith-number/MOFYXRZ76XTOK3PKSNPJA2B6F7/events.json","paper":"https://pith.science/paper/MOFYXRZ7"},"agent_actions":{"view_html":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7","download_json":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7.json","view_paper":"https://pith.science/paper/MOFYXRZ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0312010&json=true","fetch_graph":"https://pith.science/api/pith-number/MOFYXRZ76XTOK3PKSNPJA2B6F7/graph.json","fetch_events":"https://pith.science/api/pith-number/MOFYXRZ76XTOK3PKSNPJA2B6F7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7/action/storage_attestation","attest_author":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7/action/author_attestation","sign_citation":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7/action/citation_signature","submit_replication":"https://pith.science/pith/MOFYXRZ76XTOK3PKSNPJA2B6F7/action/replication_record"}},"created_at":"2026-07-04T14:59:38.910440+00:00","updated_at":"2026-07-04T14:59:38.910440+00:00"}