{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:Z4XMJOY5D2BNSEOJRIDEYIEDEF","short_pith_number":"pith:Z4XMJOY5","schema_version":"1.0","canonical_sha256":"cf2ec4bb1d1e82d911c98a064c2083217c1f28c39716ede2d803c74fa54bdaa9","source":{"kind":"arxiv","id":"1311.6364","version":2},"attestation_state":"computed","paper":{"title":"Quartic residues and sums involving $\\binom{4k}{2k}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2013-11-25T16:59:35Z","abstract_excerpt":"Let $p$ be an odd prime and let $m\\not\\equiv 0\\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\\sum_{k=0}^{[p/4]}\\binom{4k}{2k}\\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\\in\\Bbb Z$. When $p\\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\\frac{q-1}4}\\equiv (\\frac ab)^r\\pmod q$ if and only if $$\\sum_{k=0}^{[p/4]}\\binom{4k}{2k}\\Big(\\frac{a^2}{16q}\\Big)^k\\equiv (-1)^{\\frac{p^2-1}8a+\\frac{p-1}2\\cdot \\frac{q-1}4}\\Big(\\frac pq\\Big) \\Big"},"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":"1311.6364","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-25T16:59:35Z","cross_cats_sorted":[],"title_canon_sha256":"e0be7757f4b9ec0d3254764e73e675f2d7c1a4c3fec45a162db82e9f0d9f05a6","abstract_canon_sha256":"5eea8d766009ab72420ebf84143543cddc9491e0304a64b2388cc7c5e0c41f08"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:47.222664Z","signature_b64":"YKyqc6aEWa2Dq07nm55zj4xihmxSsbRaRtMS9udIppaICOMKM0E3sMYxKnqBrxcWbIqOeyRR2BR5yxUi9BgFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf2ec4bb1d1e82d911c98a064c2083217c1f28c39716ede2d803c74fa54bdaa9","last_reissued_at":"2026-05-18T03:05:47.222158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:47.222158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quartic residues and sums involving $\\binom{4k}{2k}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2013-11-25T16:59:35Z","abstract_excerpt":"Let $p$ be an odd prime and let $m\\not\\equiv 0\\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\\sum_{k=0}^{[p/4]}\\binom{4k}{2k}\\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\\in\\Bbb Z$. When $p\\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\\frac{q-1}4}\\equiv (\\frac ab)^r\\pmod q$ if and only if $$\\sum_{k=0}^{[p/4]}\\binom{4k}{2k}\\Big(\\frac{a^2}{16q}\\Big)^k\\equiv (-1)^{\\frac{p^2-1}8a+\\frac{p-1}2\\cdot \\frac{q-1}4}\\Big(\\frac pq\\Big) \\Big"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6364","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1311.6364","created_at":"2026-05-18T03:05:47.222229+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6364v2","created_at":"2026-05-18T03:05:47.222229+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6364","created_at":"2026-05-18T03:05:47.222229+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z4XMJOY5D2BN","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z4XMJOY5D2BNSEOJ","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z4XMJOY5","created_at":"2026-05-18T12:28:09.283467+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/Z4XMJOY5D2BNSEOJRIDEYIEDEF","json":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF.json","graph_json":"https://pith.science/api/pith-number/Z4XMJOY5D2BNSEOJRIDEYIEDEF/graph.json","events_json":"https://pith.science/api/pith-number/Z4XMJOY5D2BNSEOJRIDEYIEDEF/events.json","paper":"https://pith.science/paper/Z4XMJOY5"},"agent_actions":{"view_html":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF","download_json":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF.json","view_paper":"https://pith.science/paper/Z4XMJOY5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6364&json=true","fetch_graph":"https://pith.science/api/pith-number/Z4XMJOY5D2BNSEOJRIDEYIEDEF/graph.json","fetch_events":"https://pith.science/api/pith-number/Z4XMJOY5D2BNSEOJRIDEYIEDEF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF/action/storage_attestation","attest_author":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF/action/author_attestation","sign_citation":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF/action/citation_signature","submit_replication":"https://pith.science/pith/Z4XMJOY5D2BNSEOJRIDEYIEDEF/action/replication_record"}},"created_at":"2026-05-18T03:05:47.222229+00:00","updated_at":"2026-05-18T03:05:47.222229+00:00"}