{"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. M. Rains, Nadia Heninger, N. J. A. Sloane","submitted_at":"2005-09-14T14:59:50Z","abstract_excerpt":"Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f in R (where R = 1 + xZ[[x]]) can be written as f = g^n for g in R, n >= 2. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509316","kind":"arxiv","version":4},"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"}