Tootfinder

The Fractal Logic of $\Phi$-adic Recursion
Milan Rosko
https://arxiv.org/abs/2510.08934 https://arxiv.org/pdf/2510.08934

The Fractal Logic of $Φ$-adic Recursion
We establish that valid $Σ_1$ propositional inference admits reduction to Fibonacci-indexed witness equations. Specifically, modus ponens verification reduces to solving a linear Diophantine equation in $O(M(\log n))$ time, where $M$ denotes integer multiplication complexity. This reduction is transitive: tautology verification proceeds via Fibonacci index arithmetic, bypassing semantic evaluation entirely. The core discovery is a transitive closure principle in $Φ$-scaled space (Hausdorff di…

u wot m8
"""
First install the version of this crate corresponding to the Fibonacci number you want to use:
cargo add fibonacci-numbers@186 --rename fib186
Then you can use the VALUE constant to get its value:
assert_eq!(fib186::VALUE, 332825110087067562321196029789634457848);
"""
https://

this is exceedingly silly https://docs.rs/crate/fibonacci-numbers/latest/source/Cargo.toml

General Recurrence Multidimensional Zeckendorf Representations
Jiarui Cheng, Steven J. Miller, Sebastian Rodriguez-Labastida, Tianyu Shen, Alan Sun, Garrett Tresch
https://arxiv.org/abs/2510.07237

General Recurrence Multidimensional Zeckendorf Representations
We present a multidimensional generalization of Zeckendorf's Theorem (any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers) to a large family of linear recurrences. This extends work of Anderson and Bicknell-Johnson in the multi-dimensional case when the underlying recurrence is the same as the Fibonacci one. Our extension applies to linear recurrence relations defined by vectors $\vec{\mathbf{c}} = (c_1, c_2, \ldots, c_k)$ such that $c_1\geq c_2\geq\cdots \ge…

Zeros and Orthogonality of generalized Fibonacci polynomials
Cristian F. Coletti, Rigoberto Fl\'orez, Robinson A. Higuita, Sandra Z. Yepes
https://arxiv.org/abs/2510.00074 h…

Zeros and Orthogonality of generalized Fibonacci polynomials
This paper analyzes the concept of orthogonality in second-order polynomial sequences that have Binet formula similar to that of the Fibonacci and Lucas numbers, referred to as Generalized Fibonacci Polynomials (GFP). We give a technique to find roots of the GFP. As a corollary of this result, we give an alternative proof of a special case of Favard's Theorem. The general case of Favard's Theorem guarantees that there is a measure to determine whether a sequence of second-order polynomials is o…

RE: https://mastodon.acm.org/@CerstinMahlow/115673628544577463
OMG this is addictive! Just be wary that it's in KM, so my first few guesses were way off (about a pair of Fibonacci numbers off, in fact <-;).

Balanced Fibonacci word rectangles, and beyond
Jeffrey Shallit, Ingrid Vukusic
https://arxiv.org/abs/2509.25994 https://arxiv.org/pdf/2509.25994

Balanced Fibonacci word rectangles, and beyond
Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational.

Crosslisted article(s) found for cs.FL. https://arxiv.org/list/cs.FL/new
[1/1]:
- Balanced Fibonacci word rectangles, and beyond
Jeffrey Shallit, Ingrid Vukusic
h…

Crosslisted article(s) found for cs.DM. https://arxiv.org/list/cs.DM/new
[1/1]:
- Balanced Fibonacci word rectangles, and beyond
Jeffrey Shallit, Ingrid Vukusic
h…

A Note on Conjectures of Gullerud, Johnson, and Mbirika
Robert Davis, Nayda Farnsworth
https://arxiv.org/abs/2510.02008 https://arxiv.org/pdf/2510.02008

A Note on Conjectures of Gullerud, Johnson, and Mbirika
In 2023, Gullerud, Johnson, and Mbirika presented results on their study of certain tridiagonal real symmetric matrices. As part of their work, they studied the roots to nonhomogeneous equations related to characteristic polynomials of adjacency matrices for path graphs. They showed that a subset of these polynomials give a Fibonacci number when evaluated at the imaginary unit, leading them to make several intriguing conjectures. In this work, we further explore their conjectures regarding the …

The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences
Yongkang Wan, Zhonghao Liang, Qunying Liao
https://arxiv.org/abs/2509.14569

The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences
In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\leq i\leq t \right)$, Theore…

A Fibonacci-Based G\"odel Numbering: $\Delta_0$ Semantics Without Exponentiation
Milan Rosko
https://arxiv.org/abs/2509.10382 https://arxiv.org/pdf/25…

A Fibonacci-Based Gödel Numbering: $Δ_0$ Semantics Without Exponentiation
We introduce a more efficient Gödel encoding scheme based on Zeckendorf representations of natural numbers, avoiding the use of unbounded exponentiation and without appealing to the fundamental theorem of arithmetic. The resulting encoding is injective, primitive recursive, and $Δ_0$-definable in weak arithmetical theories such as $IΔ_0 + Ω_1$. This allows for the construction of fixed points and the formalization of incompleteness theorems entirely within bounded arithmetic. Compared to tr…

Conjectures About Cyclic Numbers: Resolutions and Counterexamples
Duc Hieu Le
https://arxiv.org/abs/2509.26138 https://arxiv.org/pdf/2509.26138

Conjectures About Cyclic Numbers: Resolutions and Counterexamples
We settle 22 conjectures of Cohen about cyclic numbers (positive integers $n$ with $\gcd(n,φ(n))=1$), proving 17 and disproving 5, and we completely resolve a related OEIS problem about sequences whose running averages are Fibonacci numbers. Highlights include: asymptotics for cyclics between consecutive squares with a second-order term (Conj.~9), Legendre- and $k$-fold Oppermann-type results in short quadratic intervals (Conj.~6, Conj.~20, and twin cyclics between cubes, Conj.~32), a Firoozba…

P.S. I've never even heard of then non-generalized Gaussian Fibonacci integers... I'll find out tomorrow. 🥱

Some Structures arising from the Farey Fractal
Shai Haran
https://arxiv.org/abs/2509.10959 https://arxiv.org/pdf/2509.10959

Some Structures arising from the Farey Fractal
This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the Zeckendorf shift, equipped with a metric, order, and commutative rig structure. A normalization lemma provides canonical representatives, while quotients of the non-zero Fibonadic numbers under shifts and it's fundamental domain covering the circle via phi-value…