Pith Number

pith:YOT3RY5U

pith:2026:YOT3RY5UGSQRCL4GSHNUYXRJSV

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Multiplicative independence in the sequence of $k$-generalized Pell numbers

Bernadette Faye, Cherif B. Deme, Kancou D. Fall, Khady Faye

For k at least 2, the k-generalized Pell sequence has multiplicatively dependent terms only for a short list of small indices.

arxiv:2605.17699 v1 · 2026-05-17 · math.NT

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\usepackage{pith}
\pithnumber{YOT3RY5UGSQRCL4GSHNUYXRJSV}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] A. Baker and H. Davenport,The equations3x 2−2y2 = 1and8x 2−7y2 = 1, Quarterly Journal of Mathematics 20(1969), 129–137 1969

[2] H. Batte, M. Ddamulira, J. Kasozi, and F. Luca,Multiplicative independence in the sequence ofk-generalized Lucas numbers, Indagationes Mathematicae, 2024 2024

[3] J. J. Bravo and J. L. Herrera, J.L., Repdigits in Generalized Pell Sequences,Archivum Mathematicum56 (2020), 249–262 2020

[4] J. J. Bravo, J. L. Herrera and F. Luca, On a generalization of the Pell sequence,Math. Bohem.146(2021), 199–213 2021

[5] R. D. Carmichael,On the numerical factors of the arithmetic formsα n ±β n, Annals of Mathematics15 (1913), 49–70 1913

Receipt and verification

First computed	2026-05-20T00:04:53.453217Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c3a7b8e3b434a1112f8691db4c5e2995426638a08849fa673f6e4f512dc75a1d

Aliases

arxiv: 2605.17699 · arxiv_version: 2605.17699v1 · doi: 10.48550/arxiv.2605.17699 · pith_short_12: YOT3RY5UGSQR · pith_short_16: YOT3RY5UGSQRCL4G · pith_short_8: YOT3RY5U

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YOT3RY5UGSQRCL4GSHNUYXRJSV \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c3a7b8e3b434a1112f8691db4c5e2995426638a08849fa673f6e4f512dc75a1d

Canonical record JSON

{
  "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
  }
}