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We prove that if there exist natural numbers $n$ and $d$ satisfying $n\\geq \\alpha+ d$ for which either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to $|f'(n)|$, then $f$ is irreducible in $\\mathbb{Z}[x]$."},"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":"2301.00107","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-12-31T03:19:03Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"398a7601cd69cc5462875d34dba22dab0425cf225c6d668d3b316a3b79575e20","abstract_canon_sha256":"6d7ffc34f9e3901c21d91dbaa2ee0a08d7285fc0c414ddd675cdb6cf00208890"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:29:42.686092Z","signature_b64":"/qcp2QwBIdVuMbz1Bu1Th83IRfo50iuvwFwbZYTFXrN1sX7SiCgZDA7F5mJ0EgQdxoIPSvDLsjvDSX5bQ0wrBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0368533092013d664d7aa84a999f8f4c45efe89ffac406ddc9999e8a5a2eab09","last_reissued_at":"2026-07-05T05:29:42.685526Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:29:42.685526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Another irreducibility criterion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Jitender Singh, Sanjeev Kumar","submitted_at":"2022-12-31T03:19:03Z","abstract_excerpt":"Let $f=a_0+ a_{1}x+\\cdots+a_m x^m\\in \\Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\\alpha$ such that $|a_m| \\alpha^m>|a_0|+|a_1|\\alpha+\\cdots+|a_{m-1}|\\alpha^{m-1}$. We prove that if there exist natural numbers $n$ and $d$ satisfying $n\\geq \\alpha+ d$ for which either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to $|f'(n)|$, then $f$ is irreducible in $\\mathbb{Z}[x]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.00107","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/2301.00107/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":"2301.00107","created_at":"2026-07-05T05:29:42.685594+00:00"},{"alias_kind":"arxiv_version","alias_value":"2301.00107v1","created_at":"2026-07-05T05:29:42.685594+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2301.00107","created_at":"2026-07-05T05:29:42.685594+00:00"},{"alias_kind":"pith_short_12","alias_value":"ANUFGMESAE6W","created_at":"2026-07-05T05:29:42.685594+00:00"},{"alias_kind":"pith_short_16","alias_value":"ANUFGMESAE6WMTL2","created_at":"2026-07-05T05:29:42.685594+00:00"},{"alias_kind":"pith_short_8","alias_value":"ANUFGMES","created_at":"2026-07-05T05:29:42.685594+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/ANUFGMESAE6WMTL2VBFJTH4PJR","json":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR.json","graph_json":"https://pith.science/api/pith-number/ANUFGMESAE6WMTL2VBFJTH4PJR/graph.json","events_json":"https://pith.science/api/pith-number/ANUFGMESAE6WMTL2VBFJTH4PJR/events.json","paper":"https://pith.science/paper/ANUFGMES"},"agent_actions":{"view_html":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR","download_json":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR.json","view_paper":"https://pith.science/paper/ANUFGMES","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2301.00107&json=true","fetch_graph":"https://pith.science/api/pith-number/ANUFGMESAE6WMTL2VBFJTH4PJR/graph.json","fetch_events":"https://pith.science/api/pith-number/ANUFGMESAE6WMTL2VBFJTH4PJR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR/action/storage_attestation","attest_author":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR/action/author_attestation","sign_citation":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR/action/citation_signature","submit_replication":"https://pith.science/pith/ANUFGMESAE6WMTL2VBFJTH4PJR/action/replication_record"}},"created_at":"2026-07-05T05:29:42.685594+00:00","updated_at":"2026-07-05T05:29:42.685594+00:00"}