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Let P_n := {g^n : g in R} and let mu_n := n Product_{p|n} p. We show among other things that (i) for f in R, f in P_n <=> f mod mu_n in P_n, and (ii) if f in P_n, there is a unique g in P_n with coefficients mod mu_n/n such that f == g^n (mod mu_n). In particular, if f == 1 (mod mu_n) th"},"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":"math/0509316","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-09-14T14:59:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e86377db4c33e4f47c299c4cbfa6853d403e7d1fa3bf7294825bf0dfcbad7afd","abstract_canon_sha256":"4a63d3cbc3568525e9d4d7297d9280b9d1a65aeb137863fc79f130a8d5f97f52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:39.996246Z","signature_b64":"6/wqA+XN0wLR5hvF+TodKJGey7isqqQER5mj3A5Bn/YOjsZkKFGAm7CXxkHW6D2FwfVg7395Qd/jAMxLW+1cBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db5187815d16ba88dc95d60722acc40e9b8aafdcd02b1a0ab03e61bf0291307e","last_reissued_at":"2026-05-18T02:42:39.995639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:39.995639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Integrality of n-th Roots of Generating Functions","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"E. 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