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Let $g_{n,q}\\in\\Bbb F_p[{\\tt x}]$ ($n\\ge 0$, $p=\\text{char}\\,\\Bbb F_q$) be the polynomial defined by the functional equation $\\sum_{c\\in\\Bbb F_q}({\\tt x}+c)^n=g_{n,q}({\\tt x}^q-{\\tt x})$. We determine all $n$ of the form $n=q^\\alpha-q^\\beta-1$, $\\alpha>\\beta\\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\\Bbb F_{q^2}$."},"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":"1309.3530","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-13T18:15:14Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"58ed85ab3b8617db5c83fc9bfa9937e358458d0e258e0eb98a11136453f4f4cb","abstract_canon_sha256":"a0cef578a19cc687e1ac6a76727cddb11f3d7c25aa1c8312bb748f4703adc40f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:22.304249Z","signature_b64":"TkqV3NW4PpGpMH0y21s1orc1xMLIQChtx/pWePIdCwAFlKWEfbsF6xhJaTFixJxw0JZkVg4krh0MFb66yWMNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12a3f82d75d8e328e45d47d75f99b6bf484a4e62a5f878a2cc46e3161e965d6b","last_reissued_at":"2026-05-18T03:13:22.303548Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:22.303548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Determination of a Type of Permutation Trinomials over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiang-dong Hou","submitted_at":"2013-09-13T18:15:14Z","abstract_excerpt":"Let $f=a{\\tt x} +b{\\tt x}^q+{\\tt x}^{2q-1}\\in\\Bbb F_q[{\\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\\in\\Bbb F_p[{\\tt x}]$ ($n\\ge 0$, $p=\\text{char}\\,\\Bbb F_q$) be the polynomial defined by the functional equation $\\sum_{c\\in\\Bbb F_q}({\\tt x}+c)^n=g_{n,q}({\\tt x}^q-{\\tt x})$. We determine all $n$ of the form $n=q^\\alpha-q^\\beta-1$, $\\alpha>\\beta\\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\\Bbb F_{q^2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3530","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":""},"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":"1309.3530","created_at":"2026-05-18T03:13:22.303669+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.3530v1","created_at":"2026-05-18T03:13:22.303669+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3530","created_at":"2026-05-18T03:13:22.303669+00:00"},{"alias_kind":"pith_short_12","alias_value":"CKR7QLLV3DRS","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"CKR7QLLV3DRSRZC5","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"CKR7QLLV","created_at":"2026-05-18T12:27:40.988391+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/CKR7QLLV3DRSRZC5I7LV7GNWX5","json":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5.json","graph_json":"https://pith.science/api/pith-number/CKR7QLLV3DRSRZC5I7LV7GNWX5/graph.json","events_json":"https://pith.science/api/pith-number/CKR7QLLV3DRSRZC5I7LV7GNWX5/events.json","paper":"https://pith.science/paper/CKR7QLLV"},"agent_actions":{"view_html":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5","download_json":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5.json","view_paper":"https://pith.science/paper/CKR7QLLV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.3530&json=true","fetch_graph":"https://pith.science/api/pith-number/CKR7QLLV3DRSRZC5I7LV7GNWX5/graph.json","fetch_events":"https://pith.science/api/pith-number/CKR7QLLV3DRSRZC5I7LV7GNWX5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5/action/storage_attestation","attest_author":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5/action/author_attestation","sign_citation":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5/action/citation_signature","submit_replication":"https://pith.science/pith/CKR7QLLV3DRSRZC5I7LV7GNWX5/action/replication_record"}},"created_at":"2026-05-18T03:13:22.303669+00:00","updated_at":"2026-05-18T03:13:22.303669+00:00"}