Cofactor Dynamics and Pell Equations in GCD-Augmented Fibonacci Recurrences: The Algebraic Structure of Geometric Stabilization
Mosab Hawarey
Director, Geospatial Research
Correction Notice
Issued: 2026 October 09 | Updated: 2026 October 21
The following corrections apply to this article.
- Corollary 5 (also Section 6.1): (1, 2) and (3, 5) ↔ (5, 9) are the only coprime periodic cofactor orbits of period 1 or 2, but not in general. Under the dynamics of Lemma 4, (29, 47) → (47, 77) → (77, 125) → (125, 203) is a coprime orbit of period 4 (multiplier 7), and orbits of longer period also exist.
- Theorem 10, case (III), and Section 6.3: odd seeds divisible by 3 can still reach a periodic orbit. The seeds p = 3 and p = 21 reach the (3, 5) orbit (at n = 2 and n = 7), and p = 9, 15, 27, 33 and 39 reach the period-4 orbit above.
- Section 4.2: a = (x − 3)/5 is an integer exactly when n is odd, as used in the proof of Theorem 8, rather than when n ≡ 0 (mod 3).
- Table 2: the columns labelled C(5) and C(6) list the values of C(6) and C(7).
- Section 6.2: of the two attractor pairs, only (3, 5) consists of consecutive Leonardo numbers.
The cofactor factorization (Theorem 5), the uniqueness results for period-1 and period-2 orbits (Theorems 6–8), Theorem 9 and cases (I)–(II) of Theorem 10 are unaffected.
A complete algebraic proof of the period-1 and period-2 results, and a classification of all orbits with a single amplification, are given in: Hawarey, M. (2026). Exactly Three Single-Amplification Cycles in GCD-Augmented Fibonacci Recurrences. AIR Journal of Mathematics and Computational Sciences, Vol. 2026, AIRMCS2026962. https://doi.org/10.65737/AIRMCS2026962
Abstract
We investigate the algebraic structure underlying geometric stabilization in GCD-augmented Fibonacci recurrences Cp(n) = Cp(n−1) + Cp(n−2) + gcd(Cp(n−1), Cp(n−2)), resolving Open Problem 4 from Hawarey (2026). We establish that the binary operation f(a,b) = a + b + gcd(a,b) admits the cofactor factorization f(a,b) = gcd(a,b) · (α + β + 1), which decomposes the recurrence into scale factor dynamics and a cofactor map T(α,β) = (β, α+β+1). When consecutive terms are coprime, this map coincides with the Leonardo number recurrence (ratio φ). We prove that the cofactor pair (1,2) is the unique period-1 orbit (scaling factor 2, governing even seeds) and that (3,5) is the unique coprime period-2 orbit (scaling factor 3, governing odd seeds coprime to 3), by reducing the existence condition to the generalized Pell equation x² − 5y² = 4 — the norm equation in the Fibonacci field Q(√5). The discriminant 5 thus plays a dual role: producing the irrational golden ratio φ = (1+√5)/2 in classical Fibonacci theory and the rational geometric ratio 3 in GCD-Fibonacci theory, through different representations of the same algebraic number field. We formalize the phase transition from Leonardo dynamics to geometric lock-in and provide a unified classification of all three behavioral regimes. All results are verified computationally for seeds p = 1 through 49.
Keywords
2020 Mathematics Subject Classification
How to Cite
APA:
Hawarey, M. (2026). Cofactor Dynamics and Pell Equations in GCD-Augmented Fibonacci Recurrences: The Algebraic Structure of Geometric Stabilization. AIR Journal of Mathematics and Computational Sciences, Vol. 2026, AIRMCS2026571.
https://doi.org/10.65737/AIRMCS2026571
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Copyright & Open Access
© 2026 Mosab Hawarey. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Authors retain full copyright to their work.
