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We prove that $\\varphi(q,p(x))|(q^{{\\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \\; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducib"},"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":"1312.3107","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-11T10:35:01Z","cross_cats_sorted":[],"title_canon_sha256":"16077b5643bb5b08d786f519ca02973a51c3e5862da0afdf14f6ff7694656702","abstract_canon_sha256":"90f091539723ff34d5da6957ef85cba49508d701d258255ac18acb92ec6aaa6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:58.388565Z","signature_b64":"0BTpsjY6+1tlSXJUPiCqnvXPhAfduXGC63hRzrk0BUDsYriui8ab1m4lFpu4yYSIi0SkkUOMk2Ld1QGqqKE0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83cd587164867135fcd4e9e1bdf60760919accb1c1adca8755bb5d22a5e36e8a","last_reissued_at":"2026-05-18T00:54:58.388150Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:58.388150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lehmer's totient problem over $\\mathbb{F}_q[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hourong Qin, Qingzhong Ji","submitted_at":"2013-12-11T10:35:01Z","abstract_excerpt":"In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\\in\\mathbb{F}_q[x]$ and $\\varphi(q,p(x))$ be the Euler's totient function of $p(x)$ over $\\mathbb{F}_q[x],$ where $\\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $\\varphi(q,p(x))|(q^{{\\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \\; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducib"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3107","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":""},"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":"1312.3107","created_at":"2026-05-18T00:54:58.388205+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.3107v3","created_at":"2026-05-18T00:54:58.388205+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3107","created_at":"2026-05-18T00:54:58.388205+00:00"},{"alias_kind":"pith_short_12","alias_value":"QPGVQ4LEQZYT","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"QPGVQ4LEQZYTL7GU","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"QPGVQ4LE","created_at":"2026-05-18T12:27:57.521954+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/QPGVQ4LEQZYTL7GU5HQ335QHMC","json":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC.json","graph_json":"https://pith.science/api/pith-number/QPGVQ4LEQZYTL7GU5HQ335QHMC/graph.json","events_json":"https://pith.science/api/pith-number/QPGVQ4LEQZYTL7GU5HQ335QHMC/events.json","paper":"https://pith.science/paper/QPGVQ4LE"},"agent_actions":{"view_html":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC","download_json":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC.json","view_paper":"https://pith.science/paper/QPGVQ4LE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.3107&json=true","fetch_graph":"https://pith.science/api/pith-number/QPGVQ4LEQZYTL7GU5HQ335QHMC/graph.json","fetch_events":"https://pith.science/api/pith-number/QPGVQ4LEQZYTL7GU5HQ335QHMC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC/action/storage_attestation","attest_author":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC/action/author_attestation","sign_citation":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC/action/citation_signature","submit_replication":"https://pith.science/pith/QPGVQ4LEQZYTL7GU5HQ335QHMC/action/replication_record"}},"created_at":"2026-05-18T00:54:58.388205+00:00","updated_at":"2026-05-18T00:54:58.388205+00:00"}