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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"},"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":"2605.24909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","cross_cats_sorted":[],"title_canon_sha256":"f004434d35b0899dc8a81511215ba05c277ba5795977d79362c4286fa29488c5","abstract_canon_sha256":"a9163815129559208d2e7e082f10b423f59f967c94b4d8156337111b624283c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:04:05.184177Z","signature_b64":"1GLT3nJvHe1Vww3tbOY3BnmcwB2kYXnEo38+o4JUMEXeENgttQGfqMR6pkMWFUqneIZBZBI6G7ePBMMX9ZqfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","last_reissued_at":"2026-05-26T01:04:05.183393Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:04:05.183393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $A$. 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