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bibkey: byrapuram2024pellpartialsums authors: Nikhil Byrapuram; Adam Ge; Selena Ge; Tanya Khovanova; Sylvia Zia Lee; Rajarshi Mandal; Gordon Redwine; Soham Samanta; Daniel Wu; Danyang Xu; Ray Zhao year: 2024 title: “Fibonacci Partial Sums Tricks” doi: 10.1080/00150517.2025.2556152 url: https://arxiv.org/html/2409.01296v1 claim: “Conjecture 18: For the Pell sequence, we have m^P_{4k+1}=m^P_{4k+2}=1, m^P_{4k-1}=2k, and m^P_{4k}=2k+1.” strata_touched:

  • D5/S1/Recurrence/PellPartialSumMaxIndex license: citation-only triage: anchor

Pell partial sums and their greatest dividing indices

Byrapuram et al., Fibonacci Partial Sums Tricks, arXiv:2409.01296v1 (2 September 2024), Section 4.5, states:

Conjecture 18. For the Pell sequence, we have m^P_{4k+1}=m^P_{4k+2}=1, m^P_{4k-1}=2k, and m^P_{4k}=2k+1.

The paper defines the Pell sequence by P(0)=0, P(1)=1 and P(n+2)=2P(n+1)+P(n). Its notation S_n^P is the sum of the first n terms starting at index one, and m_n^P is the greatest positive index whose Pell term divides that sum. The classes 4k+1 and 4k+2 use all k>=0; the classes 4k-1 and 4k use k>=1, so their sums are nonempty. The formal statement uses IsGreatest and the existing Pell definitions in D5/S1/Recurrence/PellCompanionGcd.

The journal DOI is 10.1080/00150517.2025.2556152, in The Fibonacci Quarterly. OpenAlex reports its online publication date as 24 June 2026. The arXiv API returns v1 as the current version. The HTML source contains Conjecture 18 in Section 4.5 and cites Bradie 2010 and Falcón Santana–Díaz-Barrero 2006 for the two even-class partial-sum identities.

Bradie, Extensions and Refinements of Some Properties of Sums Involving Pell Numbers, Missouri Journal of Mathematical Sciences 22(1) (2010), 37–43, DOI 10.35834/mjms/1312232719, has an abstract describing square characterizations of initial sums and divisibility properties. Its full text was not accessible in the checked endpoints; the abstract does not resolve the greatest-index question.

Verified locator

  • https://arxiv.org/html/2409.01296v1 : Section 4.5, Conjecture 18, sequence convention and references; retrieved 2026-10-03.
  • https://export.arxiv.org/api/query?id_list=2409.01296 : current entry 2409.01296v1; retrieved 2026-10-03.
  • https://doi.org/10.1080/00150517.2025.2556152 : journal DOI; bibliographic metadata checked through OpenAlex on 2026-10-03.
  • https://api.openalex.org/works/https://doi.org/10.1080/00150517.2025.2556152 : work W4402954534, cited-by count 0; retrieved 2026-10-03.
  • https://api.openalex.org/works?filter=cites:W4402954534 : zero indexed citing works; retrieved 2026-10-03. This is an index result, not an exhaustive assertion about all subsequent literature.
  • https://api.openalex.org/works/https://doi.org/10.35834/mjms/1312232719 : Bradie metadata and abstract; retrieved 2026-10-03.

The bounded prior-work check in issue #12462 and these sources do not establish publication priority or global absence of a prior resolution.