bibkey: mbirikaspilker2022gibonaccipowers
authors: aBa Mbirika; Jürgen Spilker
year: 2022
title: "GCD of sums of k consecutive squares of generalized Fibonacci numbers"
doi: 10.1080/00150517.2022.12427444
url: https://arxiv.org/abs/2204.12626v2
claim: "Question 6.2 asks whether dividing integer Gibonacci seeds by their gcd divides the gcd of all shifted e-th-power windows by the e-th power of that seed gcd, for every e >= 3."
strata_touched:
  - D5/S1/Recurrence/GibonacciPowerGcdNormalization
license: citation-only
triage: anchor

Integer seed content and Gibonacci power-window gcds

The article appears in Fibonacci Quarterly 60(5) (2022), 255–269. The version used here is arXiv:2204.12626v2, 13 January 2023. Definition 2.1, PDF p. 3, permits arbitrary integer initial values and uses the recurrence G(n+2) = G(n+1) + G(n). The introduction, pp. 1–2, defines the gcd over every natural shift of a finite window of positive length, rather than over a finite prefix of shifts.

Section 6.1, Question 6.2, PDF p. 14, states:

Fix $G_0,G_1\in \mathbb{Z}$ and set $d:=\gcd (G_0,G_1)$. Consider the two Gibonacci sequences $(G_n{)}_{n\ge 0}$ and $(G_n^{\prime }{)}_{n\ge 0}$, where $(G_n^{\prime }{)}_{n=0}^{\infty }$ is generated by the relatively prime initial conditions $G_0^{\prime }=\frac{G_0}{d}$ and $G_1^{\prime }=\frac{G_1}{d}$. Then the following identities holds:

${\mathcal{G}}_{G_0,G_1}(k)=d\cdot {\mathcal{G}}_{G_0^{\prime },G_1^{\prime }}(k)$; ${\mathcal{G}}_{G_0,G_1}^2(k)=d^2\cdot {\mathcal{G}}_{G_0^{\prime },G_1^{\prime }}^2(k)$.

The first formula holds from Guyer-Mbirika [4, Theorem 19], while the second formula holds from Theorem 3.6 in this current paper. Will it be the case that ${\mathcal{G}}_{G_0,G_1}^n(k)=d^n\cdot {\mathcal{G}}_{G_0^{\prime },G_1^{\prime }}^n(k)$ for all $n\ge 3$?

The quotients are defined on the nonzero-pair domain. Signed seeds, either seed individually zero, all positive window lengths, all natural exponents at least three including odd exponents, and all natural shifts are included. Both gcds are nonnegative and characterized by divisibility for every integer divisor; zero-valued windows are retained.

The formal argument combines linearity of the seeded recurrence with the universal property of Mathlib’s gcd of a set of natural absolute values. Those general algebraic facts are distinct from a source-published answer to Question 6.2. The inspected later papers arXiv:2301.05758v3 and arXiv:2408.14632v2, and the targeted source and author searches, contained no whole-question answer in the inspected passages. This bounded search does not establish global openness or literature priority. The Spilker 2016 chapter cited by related literature was not inspected.

Verified locator

• DOI: 10.1080/00150517.2022.12427444
• URL: https://arxiv.org/abs/2204.12626v2
• URL: https://arxiv.org/pdf/2204.12626v2
• Locator: Definition 2.1, PDF p. 3; Section 6.1, Question 6.2, PDF p. 14.
• Versioned primary PDF SHA-256: 09166fb6caa2d89d68f612bcab335636cc162b380ef1262395ff2fc1f63b94da.