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In 2002, Pouyanne proved that $\\rho(n,m)n^{1-\\frac{\\phi(m)}{m}}\\sim \\kappa_m$ where $\\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \\geqslant m$, \\[C(m) \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1} \\leqslant \\rho(n,m) \\leqslant \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1}\\] where $\\phi$ is Euler's totient function."},"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":"1807.10450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-27T06:14:47Z","cross_cats_sorted":[],"title_canon_sha256":"1da7b2057c7273f4e626fc3d466f2ca66636cb921d8c144405e83adf6856ebd6","abstract_canon_sha256":"05164db87275bd082c66c356a08fd0f2115d83166062f9986101297f7554093f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:17.231943Z","signature_b64":"ocHPvwBaY3Kt7SGBjsqt2xStigomu8O/4ORvJ2RdQ6wWzP42LFU589jvLkwXXrx6gmQSKTRcEOlQXbZSTWPFDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0d356a6c2e4a9441f4651dd68d68f61df2053926ca57f9481c826ae45619635","last_reissued_at":"2026-05-17T23:48:17.231252Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:17.231252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutations with orders coprime to a given integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheryl E. 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We show that there exists a positive constant $C(m)$ such that, for all $n \\geqslant m$, \\[C(m) \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1} \\leqslant \\rho(n,m) \\leqslant \\left(\\frac{n}{m}\\right)^{\\frac{\\phi(m)}{m}-1}\\] where $\\phi$ is Euler's totient function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10450","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":"1807.10450","created_at":"2026-05-17T23:48:17.231367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.10450v2","created_at":"2026-05-17T23:48:17.231367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10450","created_at":"2026-05-17T23:48:17.231367+00:00"},{"alias_kind":"pith_short_12","alias_value":"6DJVNJWC4SUU","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6DJVNJWC4SUUIH2G","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6DJVNJWC","created_at":"2026-05-18T12:32:08.215937+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/6DJVNJWC4SUUIH2GKHOWRVUPMH","json":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH.json","graph_json":"https://pith.science/api/pith-number/6DJVNJWC4SUUIH2GKHOWRVUPMH/graph.json","events_json":"https://pith.science/api/pith-number/6DJVNJWC4SUUIH2GKHOWRVUPMH/events.json","paper":"https://pith.science/paper/6DJVNJWC"},"agent_actions":{"view_html":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH","download_json":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH.json","view_paper":"https://pith.science/paper/6DJVNJWC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.10450&json=true","fetch_graph":"https://pith.science/api/pith-number/6DJVNJWC4SUUIH2GKHOWRVUPMH/graph.json","fetch_events":"https://pith.science/api/pith-number/6DJVNJWC4SUUIH2GKHOWRVUPMH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/action/storage_attestation","attest_author":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/action/author_attestation","sign_citation":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/action/citation_signature","submit_replication":"https://pith.science/pith/6DJVNJWC4SUUIH2GKHOWRVUPMH/action/replication_record"}},"created_at":"2026-05-17T23:48:17.231367+00:00","updated_at":"2026-05-17T23:48:17.231367+00:00"}