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We prove that for every finite sequence of integers $n_1,\\ldots,n_r$ there exists a divisor $f(x)=\\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\\in \\mathbb{N}$ such that $c_i=n_i$ for $1\\leq i \\leq r$. Let $H(r,n)$ denote the maximum absolute value of $r$th coefficient of divisors of $x^n-1$. In the last section of the paper we"},"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":"1511.03226","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-10T18:54:39Z","cross_cats_sorted":[],"title_canon_sha256":"6bcd38cf9e8321dc6b6af456eb3888e3d691c1d5d9d02400f7ebe3550b5fe3b5","abstract_canon_sha256":"7fe50f5024f42c6ffa2e04e938f483a1a012352ce1b5b491e7df4e43324514a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:07.162955Z","signature_b64":"oNVdtooQ8JCiqEGSdCnyvkfDKFIXQVXywj803jxfOxTHUCFYY1L01cI5FXzzNWhV1SUINA8uhXBKBYWuFGjwDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0991154a31fc184396192d9b795fe6f2c6d1234a06d54322e0d1ced8afc2af7","last_reissued_at":"2026-05-18T01:27:07.162014Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:07.162014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the coefficients of divisors of $x^n-1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sai Teja Somu","submitted_at":"2015-11-10T18:54:39Z","abstract_excerpt":"Let $a(r,n)$ be $r$th coefficient of $n$th cyclotomic polynomial. Suzuki proved that $\\{a(r,n)|r\\geq 1,n\\geq 1\\}=\\mathbb{Z}$. If $m$ and $n$ are two natural numbers we prove an analogue of Suzuki's theorem for divisors of $x^n-1$ with exactly $m$ irreducible factors. We prove that for every finite sequence of integers $n_1,\\ldots,n_r$ there exists a divisor $f(x)=\\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\\in \\mathbb{N}$ such that $c_i=n_i$ for $1\\leq i \\leq r$. Let $H(r,n)$ denote the maximum absolute value of $r$th coefficient of divisors of $x^n-1$. In the last section of the paper we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03226","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":"1511.03226","created_at":"2026-05-18T01:27:07.162128+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.03226v2","created_at":"2026-05-18T01:27:07.162128+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.03226","created_at":"2026-05-18T01:27:07.162128+00:00"},{"alias_kind":"pith_short_12","alias_value":"2CMRCVFDD7AY","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"2CMRCVFDD7AYIOLB","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"2CMRCVFD","created_at":"2026-05-18T12:28:59.999130+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/2CMRCVFDD7AYIOLBSLM3PFP6N4","json":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4.json","graph_json":"https://pith.science/api/pith-number/2CMRCVFDD7AYIOLBSLM3PFP6N4/graph.json","events_json":"https://pith.science/api/pith-number/2CMRCVFDD7AYIOLBSLM3PFP6N4/events.json","paper":"https://pith.science/paper/2CMRCVFD"},"agent_actions":{"view_html":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4","download_json":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4.json","view_paper":"https://pith.science/paper/2CMRCVFD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.03226&json=true","fetch_graph":"https://pith.science/api/pith-number/2CMRCVFDD7AYIOLBSLM3PFP6N4/graph.json","fetch_events":"https://pith.science/api/pith-number/2CMRCVFDD7AYIOLBSLM3PFP6N4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4/action/storage_attestation","attest_author":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4/action/author_attestation","sign_citation":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4/action/citation_signature","submit_replication":"https://pith.science/pith/2CMRCVFDD7AYIOLBSLM3PFP6N4/action/replication_record"}},"created_at":"2026-05-18T01:27:07.162128+00:00","updated_at":"2026-05-18T01:27:07.162128+00:00"}