{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:L727SU7BQMGU3KUA4P3BBOC2UV","short_pith_number":"pith:L727SU7B","schema_version":"1.0","canonical_sha256":"5ff5f953e1830d4daa80e3f610b85aa5675db92b29fe818a48f375c88b13317c","source":{"kind":"arxiv","id":"1709.05540","version":2},"attestation_state":"computed","paper":{"title":"Primitive Element Pairs with One Prescribed Trace over a Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Anju Gupta, R. K. Sharma, Stephen D. Cohen","submitted_at":"2017-09-16T16:58:16Z","abstract_excerpt":"In this article, we establish a sufficient condition for the existence of a primitive element $\\alpha \\in {\\mathbb{F}_{q^n}}$ such that the element $\\alpha+\\alpha^{-1}$ is also a primitive element of ${\\mathbb{F}_{q^n}},$ and $Tr_{\\mathbb{F}_{q^n}|\\mathbb{F}_{q}}(\\alpha)=a$ for any prescribed $a \\in \\mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\\mathbb{F}_{q^n}~ (n \\geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $"},"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":"1709.05540","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-09-16T16:58:16Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"3fd6c9f1d45ff2c1ec9a001a67ec9434403078348469b4e95088ecb19e84943b","abstract_canon_sha256":"9f15b328c74b9d5e1dd3b18dda9005b048c48fca3b8a58dc844273c296a35ead"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:58.948312Z","signature_b64":"NPY7AdZrmZaHZCUnKVCk4xNalBF+8kiJJcaGAowTdXfJAV/a8H9yUYBWKBTRDAEbH6Mr6dsCimnU45vyk02KCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ff5f953e1830d4daa80e3f610b85aa5675db92b29fe818a48f375c88b13317c","last_reissued_at":"2026-05-18T00:19:58.947800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:58.947800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Primitive Element Pairs with One Prescribed Trace over a Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Anju Gupta, R. K. Sharma, Stephen D. Cohen","submitted_at":"2017-09-16T16:58:16Z","abstract_excerpt":"In this article, we establish a sufficient condition for the existence of a primitive element $\\alpha \\in {\\mathbb{F}_{q^n}}$ such that the element $\\alpha+\\alpha^{-1}$ is also a primitive element of ${\\mathbb{F}_{q^n}},$ and $Tr_{\\mathbb{F}_{q^n}|\\mathbb{F}_{q}}(\\alpha)=a$ for any prescribed $a \\in \\mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\\mathbb{F}_{q^n}~ (n \\geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05540","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":"1709.05540","created_at":"2026-05-18T00:19:58.947876+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.05540v2","created_at":"2026-05-18T00:19:58.947876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05540","created_at":"2026-05-18T00:19:58.947876+00:00"},{"alias_kind":"pith_short_12","alias_value":"L727SU7BQMGU","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"L727SU7BQMGU3KUA","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"L727SU7B","created_at":"2026-05-18T12:31:28.150371+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/L727SU7BQMGU3KUA4P3BBOC2UV","json":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV.json","graph_json":"https://pith.science/api/pith-number/L727SU7BQMGU3KUA4P3BBOC2UV/graph.json","events_json":"https://pith.science/api/pith-number/L727SU7BQMGU3KUA4P3BBOC2UV/events.json","paper":"https://pith.science/paper/L727SU7B"},"agent_actions":{"view_html":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV","download_json":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV.json","view_paper":"https://pith.science/paper/L727SU7B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.05540&json=true","fetch_graph":"https://pith.science/api/pith-number/L727SU7BQMGU3KUA4P3BBOC2UV/graph.json","fetch_events":"https://pith.science/api/pith-number/L727SU7BQMGU3KUA4P3BBOC2UV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV/action/storage_attestation","attest_author":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV/action/author_attestation","sign_citation":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV/action/citation_signature","submit_replication":"https://pith.science/pith/L727SU7BQMGU3KUA4P3BBOC2UV/action/replication_record"}},"created_at":"2026-05-18T00:19:58.947876+00:00","updated_at":"2026-05-18T00:19:58.947876+00:00"}