{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:5CD6KSKE236JEEDWCJJ5QWQY7I","short_pith_number":"pith:5CD6KSKE","schema_version":"1.0","canonical_sha256":"e887e54944d6fc9210761253d85a18fa23faa4f3f6066010e5d01471adebdbbf","source":{"kind":"arxiv","id":"0712.2365","version":2},"attestation_state":"computed","paper":{"title":"Ternary cyclotomic polynomials having a large coefficient","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pieter Moree, Yves Gallot","submitted_at":"2007-12-14T15:05:49Z","abstract_excerpt":"Let $\\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\\Phi_n(x)$, satisfies $|a_n(k)|\\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\\ge 11$. We also prove that given any $\\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes"},"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":"0712.2365","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-12-14T15:05:49Z","cross_cats_sorted":[],"title_canon_sha256":"635d53d7a5d6d3a5696dba5e8c89c28e33b6adfb313392a9a003ca386b43dae5","abstract_canon_sha256":"655f2d4a8d8bc2c837376a96aa404b6cd209b7116cb76ec18bd5f3fb89fdc4bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:00.211471Z","signature_b64":"HHPEp2rR3RIWZy3seOlGXAxKzNhAP91+Nbnd1xcR+UJwO5EQVTtCKvEgeAB+ST5wshBJsN0iQVLtP7ptad/TCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e887e54944d6fc9210761253d85a18fa23faa4f3f6066010e5d01471adebdbbf","last_reissued_at":"2026-05-18T03:50:00.210914Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:00.210914Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ternary cyclotomic polynomials having a large coefficient","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pieter Moree, Yves Gallot","submitted_at":"2007-12-14T15:05:49Z","abstract_excerpt":"Let $\\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\\Phi_n(x)$, satisfies $|a_n(k)|\\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\\ge 11$. We also prove that given any $\\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2365","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":"0712.2365","created_at":"2026-05-18T03:50:00.211008+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.2365v2","created_at":"2026-05-18T03:50:00.211008+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2365","created_at":"2026-05-18T03:50:00.211008+00:00"},{"alias_kind":"pith_short_12","alias_value":"5CD6KSKE236J","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"5CD6KSKE236JEEDW","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"5CD6KSKE","created_at":"2026-05-18T12:25:55.427421+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/5CD6KSKE236JEEDWCJJ5QWQY7I","json":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I.json","graph_json":"https://pith.science/api/pith-number/5CD6KSKE236JEEDWCJJ5QWQY7I/graph.json","events_json":"https://pith.science/api/pith-number/5CD6KSKE236JEEDWCJJ5QWQY7I/events.json","paper":"https://pith.science/paper/5CD6KSKE"},"agent_actions":{"view_html":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I","download_json":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I.json","view_paper":"https://pith.science/paper/5CD6KSKE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.2365&json=true","fetch_graph":"https://pith.science/api/pith-number/5CD6KSKE236JEEDWCJJ5QWQY7I/graph.json","fetch_events":"https://pith.science/api/pith-number/5CD6KSKE236JEEDWCJJ5QWQY7I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I/action/storage_attestation","attest_author":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I/action/author_attestation","sign_citation":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I/action/citation_signature","submit_replication":"https://pith.science/pith/5CD6KSKE236JEEDWCJJ5QWQY7I/action/replication_record"}},"created_at":"2026-05-18T03:50:00.211008+00:00","updated_at":"2026-05-18T03:50:00.211008+00:00"}