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For $k\\ge 2$ we determine all pairs $(m,n)$ with $n>m\\ge 0$ such that $P_n^{(k)}$ and $P_m^{(k)}$ are multiplicatively dependent. The main result states that the only solutions occur for very small $k,m,n$ (which are listed explicitly). The proof uses lower bounds for linear forms in logarith","authors_text":"Bernadette Faye, Cherif B. Deme, Kancou D. Fall, Khady Faye","cross_cats":[],"headline":"For k at least 2, the k-generalized Pell sequence has multiplicatively dependent terms only for a short list of small indices.","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-17T23:44:44Z","title":"Multiplicative independence in the sequence of $k$-generalized Pell numbers"},"references":{"count":12,"internal_anchors":0,"resolved_work":12,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"A. Baker and H. Davenport,The equations3x 2−2y2 = 1and8x 2−7y2 = 1, Quarterly Journal of Mathematics 20(1969), 129–137","work_id":"807724d4-a76d-482b-a629-ad938835530c","year":1969},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"H. Batte, M. Ddamulira, J. Kasozi, and F. Luca,Multiplicative independence in the sequence ofk-generalized Lucas numbers, Indagationes Mathematicae, 2024","work_id":"9405d250-ee22-4939-8190-b4c6d5ce769c","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"J. J. Bravo and J. L. Herrera, J.L., Repdigits in Generalized Pell Sequences,Archivum Mathematicum56 (2020), 249–262","work_id":"bc3aca4a-6cff-40bf-a79d-d38b29b3b82f","year":2020},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"J. J. Bravo, J. L. Herrera and F. Luca, On a generalization of the Pell sequence,Math. Bohem.146(2021), 199–213","work_id":"bd2357a3-9ce2-4f24-bc2d-6ee39ec9d140","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"R. D. Carmichael,On the numerical factors of the arithmetic formsα n ±β n, Annals of Mathematics15 (1913), 49–70","work_id":"ed54e069-0e98-4ac1-bf87-9a4de6decede","year":1913}],"snapshot_sha256":"cd96ca11339e07836e5247056b52a24fe2488b54f065eb84b2bb37a13d48af4d"},"source":{"id":"2605.17699","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T21:56:11.963897Z","id":"d8a9f7a3-bd6c-4238-8ebd-204a22e15a5d","model_set":{"reader":"grok-4.3"},"one_line_summary":"For the k-generalized Pell sequence defined by the given recurrence, the only multiplicatively dependent pairs P_n^(k) and P_m^(k) occur for small listed values of k, m, n.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"For k at least 2, the k-generalized Pell sequence has multiplicatively dependent terms only for a short list of small indices.","strongest_claim":"For k ≥ 2 the only solutions with n > m ≥ 0 such that P_n^(k) and P_m^(k) are multiplicatively dependent occur for very small k, m, n which are listed explicitly.","weakest_assumption":"That the combination of Matveev's lower bounds and the Baker-Davenport reduction produces an explicit finite bound small enough for exhaustive computational verification, with no missed large solutions outside the reduced range."}},"verdict_id":"d8a9f7a3-bd6c-4238-8ebd-204a22e15a5d"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6afb669aad13448289a89eecc21ebc1efec9f45dcb5f77a38834633a1c3f2260","target":"record","created_at":"2026-05-20T00:04:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"97c6d1f62c7fbb61d9318a889be8ca6b914366a5e8e2126290eeac8d6d7494ed","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-17T23:44:44Z","title_canon_sha256":"e83077717febe8576e8212da22e99007067bcec5262cb03288d8c8e45c5ccdd8"},"schema_version":"1.0","source":{"id":"2605.17699","kind":"arxiv","version":1}},"canonical_sha256":"c3a7b8e3b434a1112f8691db4c5e2995426638a08849fa673f6e4f512dc75a1d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3a7b8e3b434a1112f8691db4c5e2995426638a08849fa673f6e4f512dc75a1d","first_computed_at":"2026-05-20T00:04:53.453217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:53.453217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/IXTD+SYYA/E9/ueKoeh/L6JPrFPL4szj9NWA2cqWYKG+cf4Qm4klJozTeYpXBHXEuVctzLYijlUDb8IndxUAQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:53.454039Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17699","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6afb669aad13448289a89eecc21ebc1efec9f45dcb5f77a38834633a1c3f2260","sha256:63f78b57751f0a5f1946962aa1acca4bdb08162b2368edd07a1fd4026b3e0a7e"],"state_sha256":"3644030d3f6a7ddcf9a3fe14a85302cf42cdea5fccbb2b46d36b20ae931b3ef6"}