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Here we show how to find the length of those cycles, in terms of $p$ and $\\xi$, using cyclotomic polynomials over $\\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $\\xi$ such that the permutation $\\tau_{\\xi,p}"},"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":"2504.08709","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-04-11T17:20:19Z","cross_cats_sorted":[],"title_canon_sha256":"1f3cb2ec87482471d1bffce32215006ee8ec4286e9e0a1e470ce6081a6fd7b3e","abstract_canon_sha256":"1f6cb943308553c205c97f7985caa13e228157e50b8e1f8654e34e0a97a8bc53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:47:53.000495Z","signature_b64":"3kTmuMZbVQr/wRyh0hjrkVUFE7SwYgWkm2J0GZ5E4A2YjmDcJhDcFXiy5sJD4Ar3JchHipRrwbm+JJ9mMBCUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a10dc0eab93b66dbbd836a5c2d60ee0809b767fcf9e72532ce8d839ff0185de0","last_reissued_at":"2026-07-05T10:47:53.000044Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:47:53.000044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Cycle Structure of the Metacommutation Map","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ant\\'onio Leite, Ant\\'onio Machiavelo","submitted_at":"2025-04-11T17:20:19Z","abstract_excerpt":"Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $\\xi$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $\\tau_{\\xi,p}$ induced on the non-fixed points splits into cycles of the same length. Here we show how to find the length of those cycles, in terms of $p$ and $\\xi$, using cyclotomic polynomials over $\\mathbb{F}_p$. 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