{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:VFENLUD7T5SURF5S5454R7EVUX","short_pith_number":"pith:VFENLUD7","schema_version":"1.0","canonical_sha256":"a948d5d07f9f654897b2ef3bc8fc95a5c4195c7a55c5f268c7a29623a0587bec","source":{"kind":"arxiv","id":"2006.14716","version":1},"attestation_state":"computed","paper":{"title":"Generalized Paley graphs and their complete subgraphs of orders three and four","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Dermot McCarthy, Madeline Locus Dawsey","submitted_at":"2020-06-25T22:13:43Z","abstract_excerpt":"Let $k \\geq 2$ be an integer. Let $q$ be a prime power such that $q \\equiv 1 \\pmod {k}$ if $q$ is even, or, $q \\equiv 1 \\pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set $\\mathbb{F}_q$ where $ab$ is an edge if and only if ${a-b}$ is a $k$-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $G_k(q)$, $\\mathcal{K}_4(G_k(q))$, which holds for all $k$. This generalizes the results of Evans, Pulham and Sheehan on the original ($k$=2) Paley "},"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":"2006.14716","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-06-25T22:13:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e53ac508a8a561024dc59bbd3b50ef42f7fdb1ca19f113b6ac34109d42d3115f","abstract_canon_sha256":"5a43d567050c7d8aa17e3c77ccf0b70c45d54012d1ac9bdbdddd6b1999b8e619"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:32:47.567388Z","signature_b64":"OHMt83DJualWl2AMv9fOYfphYYvkKGqSIyACgWZvCueHv3G9N0UNNRTCkCvJgqEhQ3oz3FSOzeWDwa3+hDlkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a948d5d07f9f654897b2ef3bc8fc95a5c4195c7a55c5f268c7a29623a0587bec","last_reissued_at":"2026-07-05T04:32:47.566758Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:32:47.566758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Paley graphs and their complete subgraphs of orders three and four","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Dermot McCarthy, Madeline Locus Dawsey","submitted_at":"2020-06-25T22:13:43Z","abstract_excerpt":"Let $k \\geq 2$ be an integer. Let $q$ be a prime power such that $q \\equiv 1 \\pmod {k}$ if $q$ is even, or, $q \\equiv 1 \\pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set $\\mathbb{F}_q$ where $ab$ is an edge if and only if ${a-b}$ is a $k$-th power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $G_k(q)$, $\\mathcal{K}_4(G_k(q))$, which holds for all $k$. This generalizes the results of Evans, Pulham and Sheehan on the original ($k$=2) Paley "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.14716","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/2006.14716/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":"2006.14716","created_at":"2026-07-05T04:32:47.566826+00:00"},{"alias_kind":"arxiv_version","alias_value":"2006.14716v1","created_at":"2026-07-05T04:32:47.566826+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.14716","created_at":"2026-07-05T04:32:47.566826+00:00"},{"alias_kind":"pith_short_12","alias_value":"VFENLUD7T5SU","created_at":"2026-07-05T04:32:47.566826+00:00"},{"alias_kind":"pith_short_16","alias_value":"VFENLUD7T5SURF5S","created_at":"2026-07-05T04:32:47.566826+00:00"},{"alias_kind":"pith_short_8","alias_value":"VFENLUD7","created_at":"2026-07-05T04:32:47.566826+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/VFENLUD7T5SURF5S5454R7EVUX","json":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX.json","graph_json":"https://pith.science/api/pith-number/VFENLUD7T5SURF5S5454R7EVUX/graph.json","events_json":"https://pith.science/api/pith-number/VFENLUD7T5SURF5S5454R7EVUX/events.json","paper":"https://pith.science/paper/VFENLUD7"},"agent_actions":{"view_html":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX","download_json":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX.json","view_paper":"https://pith.science/paper/VFENLUD7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2006.14716&json=true","fetch_graph":"https://pith.science/api/pith-number/VFENLUD7T5SURF5S5454R7EVUX/graph.json","fetch_events":"https://pith.science/api/pith-number/VFENLUD7T5SURF5S5454R7EVUX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX/action/storage_attestation","attest_author":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX/action/author_attestation","sign_citation":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX/action/citation_signature","submit_replication":"https://pith.science/pith/VFENLUD7T5SURF5S5454R7EVUX/action/replication_record"}},"created_at":"2026-07-05T04:32:47.566826+00:00","updated_at":"2026-07-05T04:32:47.566826+00:00"}