{"paper":{"title":"Valuation Separation for Coprime Lucas Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dongyeon Kym","submitted_at":"2026-05-24T07:30:03Z","abstract_excerpt":"Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\\pm 1$ and discriminant $\\Delta=P^2+4Q>0$. We study Diophantine equations \\[ A y^k=\\prod_{i=1}^r U_{n_i}(P,Q), \\qquad k\\geq 2, \\] where the indices $n_1,\\ldots,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24909","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.24909/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}