{"paper":{"title":"Multiplicative independence in the sequence of $k$-generalized Pell numbers","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"For k at least 2, the k-generalized Pell sequence has multiplicatively dependent terms only for a short list of small indices.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bernadette Faye, Cherif B. Deme, Kancou D. Fall, Khady Faye","submitted_at":"2026-05-17T23:44:44Z","abstract_excerpt":"We study multiplicative dependence between terms of the $k$-generalized Pell sequence $(P_n^{(k)})_{n\\ge 2-k}$, defined by the linear recurrence \\[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \\dots + P_{n-k}^{(k)}, \\] with initial conditions $P_0^{(k)} = \\dots = P_{-(k-2)}^{(k)} = 0$ and $P_1^{(k)} = 1$. 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For k at least 2, the k-generalized Pell sequence has multiplicatively dependent terms only for a short list of small indices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fd2bf23ceb2466306a503db066b17e5808c0134d28d37605213176ebbd4f9edc"},"source":{"id":"2605.17699","kind":"arxiv","version":1},"verdict":{"id":"d8a9f7a3-bd6c-4238-8ebd-204a22e15a5d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:56:11.963897Z","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.","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","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.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17699/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.410457Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:10:58.007955Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T21:49:43.943624Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T21:49:43.742374Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.516385Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T21:21:59.685057Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.424800Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"903dbb9551cf4567c0e6d82c08feef49d720fd80b555807dd6746aebdd770018"},"references":{"count":12,"sample":[{"doi":"","year":1969,"title":"A. 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