Tootfinder

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…

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…

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 <-;).

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…

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…

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…