Read any interesting papers? This is a good place to post a link and maybe a quick summary.
I like the old papers! On the 3x+1 Problem, by R. E. Crandall (1978) contains a lot of early work, including on the qn+1 problem, where he notes cycles for the 5n+1 and 181n+1 problems. He also shows that almost all start numbers for the 1093n+1 problem fail to reach 1.
Great source of survey papers and overviews: The Ultimate Challenge: The 3x+1 Problem by Jeffrey Lagarias (2011).
Terence Tao of UCLA proved that almost every start number n goes very low, in fact down to any f(n) you care to choose, such as log\ log\ log(n). The proof is way hard, but his blog posts are tutorial-ish, with something for everyone.
https://arxiv.org/pdf/2503.18299
How might one represent orbits as geodesic objects on a manifold?
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I really like the paper:
Monks, K. M. (2006). The sufficiency of arithmetic progressions for the 3x+1 conjecture. Proc. Amer. Math. Soc. , 134 (10), 2861â2872.
TLDR: it shows that proving the Collatz conjecture on any arithmetic progression A+Bn, with A,B \in \mathbb{N} and B \neq 0. In my recollection, it does so by exploring the ancestor graph and by tweaking parity vectors.
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Shalom Eliahou and Jean-Louis Verger-Gaugry
The number system in rational base 3/2 and the 3x +1 problem
Volume 363 (2025), p. 329-336
Here is how to read base-3/2 with the tiles: Six Collatz tiles - #2 by cosmo
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A bunch of Collatz papers from arXiv search
Title: [2511.10811v1] Transformers know more than they can tell -- Learning the Collatz sequence
Authors: François Charton, Ashvni Narayanan
Date: 2025-11-13
Abstract:
We investigate transformer prediction of long Collatz steps, a complex arithmetic function that maps odd integers to their distant successors in the Collatz sequence ( u_{n+1}=u_n/2 if u_n is even, u_{n+1}=(3u_n+1)/2 if u_n is odd). Model accuracy varies with the base used to encode input and output. It can be as high as 99.7\% for bases 24 and 32, and as low as 37 and 25\% for bases 11 and 3. Yet, all models, no matter the base, follow a common learning pattern. As training proceeds, they learn a sequence of classes of inputs that share the same residual modulo 2^p. Models achieve near-perfect accuracy on these classes, and less than 1\% for all other inputs. This maps to a mathematical property of Collatz sequences: the length of the loops involved in the computation of a long Collatz step can be deduced from the binary representation of its input. The learning pattern reflects the model learning to predict inputs associated with increasing loop lengths. An analysis of failure cases reveals that almost all model errors follow predictable patterns. Hallucination, a common feature of large language models, almost never happens. In over 90\% of failures, the model performs the correct calculation, but wrongly estimates loop lengths. Our observations give a full account of the algorithms learned by the models. They suggest that the difficulty of learning such complex arithmetic function lies in figuring the control structure of the computation â the length of the loops. We believe that the approach outlined here, using mathematical problems as tools for understanding, explaining, and perhaps improving language models, can be applied to a broad range of problems and bear fruitful results.
Title: [2510.11723v1] A Normality Conjecture on Rational Base Number Systems
Authors: MĂ©lodie Andrieu, Shalom Eliahou, LĂ©o Vivion
Date: 2025-10-06
Abstract:
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We conjecture that every minimal and maximal word is normal over an appropriate subalphabet. To support this conjecture, we present extensive numerical experiments that examine the richness threshold and the discrepancy of these words. We also discuss the implications that the validity of our conjecture would have for several long-standing open problems, including the existence of Z-numbers (Mahler, 1968) and Z_{p/q}-numbers (Flatto, 1992), the existence of triple expansions in rational base p/q (Akiyama, 2008), and the Collatz-inspired `4/3 problemâ (Dubickas and Mossinghoff, 2009).
Title: [2510.07530v1] A variant of Collatz's Conjecture over Binary Polynomials
Authors: Luis H. Gallardo, Olivier Rahavandrainy
Date: 2025-10-08
Abstract:
We study a natural analogue of Collatzâs Conjecture for polynomials over \mathbb{F}_2.
Title: [2510.06736v1] Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus
Authors: Christos N. Efrem
Date: 2025-10-08
Abstract:
This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex analysis, including contour-integral representations, residue calculus, and Cauchyâs integral formulas. The basic approach is inspired by Egorychevâs method, which has been used so far to establish combinatorial identities. Moreover, this work generalizes existing results and extends the methodology to multiple dimensions using several complex variables.
Title: [2509.21598v1] Bacterial Gene Regulatory Neural Network as a Biocomputing Library of Mathematical Solvers
Authors: Adrian Ratwatte, Samitha Somathilaka, Thanh Cao, Xu Li, Sasitharan Balasubramaniam
Date: 2025-09-25
Abstract:
Current biocomputing approaches predominantly rely on engineered circuits with fixed logic, offering limited stability and reliability under diverse environmental conditions. Here, we use the GRNN framework introduced in our previous work to transform bacterial gene expression dynamics into a biocomputing library of mathematical solvers. We introduce a sub-GRNN search algorithm that identifies functional subnetworks tailored to specific mathematical calculation and classification tasks by evaluating gene expression patterns across chemically encoded input conditions. Tasks include identifying Fibonacci numbers, prime numbers, multiplication, and Collatz step counts. The identified problem-specific sub-GRNNs are then assessed using gene-wise and collective perturbation, as well as Lyapunov-based stability analysis, to evaluate robustness and reliability. Our results demonstrate that native transcriptional machinery can be harnessed to perform diverse mathematical calculation and classification tasks, while maintaining computing stability and reliability.
Title: [2508.17285v1] Additive systems for $\mathbb{Z}$ are undecidable
Authors: Andrei Zabolotskii
Date: 2025-08-24
Abstract:
What are the collections of sets {A}_i\subset\mathbb{Z} such that any n\in\mathbb{Z} has exactly one representation as n=a_0+a_1+\dotsb with a_i\in{A}_i? The answer for \mathbb{N}_0 instead of \mathbb{Z} is given by a theorem of de Bruijn. We describe a family of natural candidate collections for \mathbb{Z}, which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire \mathbb{Z} is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.
Title: [2508.05713v1] Dynamical Systems with Bounded Condition and $C^{*}$-algebras
Authors: Takehiko Mori
Date: 2025-08-07
Abstract:
By introducing the totally uniqueness condition for maps, we establish a one-to-one correspondence between the family of invariant sets for the q x{+}d-function (e.g. the 3 x{+}1-map, also known as the Collatz map), where q and d are arbitrary positive odd numbers, and the family of reducing subspaces for the associated C^{*}-algebra. This extends the connection between the Collatz conjecture (also known as the 3 n{+}1-problem) and the irreducibility of its associated C^{*}-algebra. We also introduce homomorphisms between dynamical systems with bounded conditions that preserve the structures of these dynamical systems. We prove the existence of an isomorphism between their associated C^{*}-algebras is proven for every isomorphism between dynamical systems with bounded condition.
Title: [2506.21728v2] A Finite-State Symbolic Automaton Model for the Collatz Map and Its Convergence Properties
Authors: Leonard Ben Aurel Brauer
Date: 2025-06-26
Abstract:
We present a finite-state, deterministic automaton that emulates the Collatz function through digitwise transitions on base-10 representations. Each digit is represented as a symbolic triplet (r, p, c) encoding its value, the parity of the next digit, and an incoming carry propagated from the lower digit. This yields exactly 60 possible local states. The automaton applies local, parity-aware rules that collectively reconstruct the global arithmetic of the Collatz map. We show that all symbolic trajectories converge in finitely many steps to a unique terminal cycle (4, 0, 0) â (2, 0, 0) â (1, 0, 0), with all higher digit positions degenerating to the absorbing state (0, 0, 0). This collapse reveals a canonical symbolic normal form of Collatz dynamics.
In parallel, a binary view explains the dynamics as alternating bit-length growth and contraction, aligning with known heuristics for Collatz convergence. This structural perspective is further reinforced by a symbolic drift function and a ranking potential that together explain and formalize the convergence process.
Title: [2506.19115v1] A Two-Operator Calculus for Arithmetic-Progression Paths in the Collatz Graph
Authors: Sebastian Angermund
Date: 2025-06-23
Abstract:
A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single step, (ii) gives a closed formula for every set of seeds that realises a prescribed parity word, (iii) yields a one line affine invariant that forbids trajectories consisting of infinitely many odd moves, and (iv) reduces the non-trivial-cycle problem to a pair of linear congruences.
Title: [2504.13716v1] The number system in rational base $3/2$ and the $3x+1$ problem
Authors: Shalom Eliahou, Jean-Louis Verger-Gaugry
Date: 2025-04-18
Abstract:
The representation of numbers in rational base p/q was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case p/q=3/2. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz 3x+1 function. Our purpose in this note is to expose these links and motivate further research into them.
Title: [2504.07970v1] Collatz Representations With Bounded Partial Quotients
Authors: Franciszek Kobus
Date: 2025-03-21
Abstract:
We define Collatz representations for a subset of rational numbers and prove that each real number ( x \notin (-1,1) ) can be approximated arbitrarily well by rational numbers which have only ( 2 )'s and ( 1 )'s in their Collatz representation.
Title: [2503.16176v2] Nonnegative Biquadratic Tensors
Authors: Chunfeng Cui, Liqun Qi
Date: 2025-03-20
Abstract:
An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M$^+-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M^+-eigenvalue is called an M^{++}-eigenvalue. A nonnegative biquadratic tensor has at least one M^+ eigenvalue, and the largest M^+-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M^+-eigenvalues are M^{++}-eigenvalues. Although the M^+-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M^+-eigenvalue has a max-min characterization, while the smallest M^+-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M^+$-eigenvalues is proposed. Numerical results are reported.
Title: [2502.20642v2] Fixed point theorem in metric spaces and its application to the Collatz conjecture
Authors: Toshiharu Kawasaki
Date: 2025-02-28
Abstract:
In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.
Title: [2502.16743v1] Numerical verification of the Collatz conjecture for billion digit random numbers
Authors: Andreas-Stephan Elsenhans
Date: 2025-02-23
Abstract:
The Collatz conjecture, also known as the 3n+1 problem, is one of the most popular open problems in number theory. In this note, an algorithm for the verification of the Collatz conjecture is presented that works on a standard PC for numbers with up to ten billion decimal places.
Title: [2502.09917v1] Approximation of the generalized principal eigenvalue of cooperative nonlocal dispersal systems and applications
Authors: Mingxin Wang, Lei Zhang
Date: 2025-02-14
Abstract:
It is well known that, in the study of the dynamical properties of nonlinear evolution system with nonlocal dispersals, the principal eigenvalue of linearized system play an important role. However, due to lack of compactness, in order to obtain the existence of principal eigenvalue, certain additional conditions must be attached to the coefficients. In this paper, we approximate the generalized principal eigenvalue of nonlocal dispersal cooperative and irreducible system, which admits the Collatz-Wielandt characterization, by constructing the monotonic upper and lower control systems with principal eigenvalues; and show that the generalized principal eigenvalue plays the same role as the usual principal eigenvalue.
Title: [2502.00948v3] Paradoxical behavior in Collatz sequences
Authors: Olivier Rozier, Claude Terracol
Date: 2025-02-02
Abstract:
On the set of positive integers, we consider an iterated process that sends n to \frac{3n+1}{2} or to \frac{n}{2} depending on the parity of n. According to a conjecture due to Collatz, all such sequences end up in the cycle (1,2). In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting from n\geq2 is sufficient to determine its stopping time, namely, the number of iterations needed to descend below n. However, when iterating beyond the stopping time, there exist âparadoxicalâ sequences for which the first term is unexpectedly exceeded. In the present study, we show that this topic is strongly linked to the Collatz conjecture. Moreover, this non-typical behavior seems to occur finitely many times apart from the trivial cycle, thus lending support to Terrasâ conjecture.
Title: [2501.04032v2] Efficient Computation of Collatz Sequence Stopping Times: A Novel Algorithmic Approach
Authors: Eyob Solomon Getachew, Beakal Gizachew Assefa
Date: 2025-01-01
Abstract:
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute the stopping time of numbers in the Collatz sequence, achieving significant computational improvements. By leveraging structural patterns in the Collatz tree, the proposed algorithm minimizes redundant operations and optimizes computational steps. Unlike prior methods, it efficiently handles extremely large numbers without requiring advanced techniques such as memoization or parallelization. Experimental evaluations confirm computational efficiency improvements of approximately 28% over state-of-the-art methods. These findings underscore the algorithmâs scalability and robustness, providing a foundation for future large-scale verification of the conjecture and potential applications in computational mathematics.
Title: [2412.08097v2] The Structure of the Route to the Period-three Orbit in the Collatz Map
Authors: Weicheng Fu, Yisen Wang
Date: 2024-12-11
Abstract:
This study analyzes the Collatz map through nonlinear dynamics. By embedding integers in Sharkovskyâs ordering, we show that odd initial values suffice for full dynamical characterization. We introduce ``direction phasesââ to partition iterations into upward and downward phases, and derive a recursive function family parameterized by upward phase counts. Consequently, a logarithmic scaling law between iteration steps and initial values is revealed, demonstrating finite-time convergence to the period-three orbit. Moreover, we establish the equivalence of the Collatz map to a binary shift map, whose ergodicity guarantees universal convergence to attractors, providing additional support for convergence. Furthermore, we identify that basins of attraction follow power-law distributions and find that odd numbers classified by upward phases follow Gamma statistics. These results offer valuable insights into the dynamics of discrete systems and their connections to number theory.
Title: [2412.02902v1] $\left(p,q\right)$-adic Analysis and the Collatz Conjecture
Authors: Maxwell Charles Siegel
Date: 2024-12-03
Abstract:
What use can there be for a function from the p-adic numbers to the q-adic numbers, where p and q are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. â\left(p,q\right)-adic analysisâ of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map H:\mathbb{Z}\rightarrow\mathbb{Z}, one can construct a function Ï_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q} for an appropriate choice of distinct primes p,q with the property that x\in\mathbb{Z}\backslash\left\{ 0\right\} is a periodic point of H if and only if there is a p-adic integer \mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} so that Ï_{H}\left(\mathfrak{z}\right)=x. By generalizing Monna-Springer integration theory and establishing a \left(p,q\right)-adic analogue of the Wiener Tauberian Theorem, one can show that the question âis x\in\mathbb{Z}\backslash\left\{ 0\right\} a periodic point of H?â is essentially equivalent to âis the span of the translates of the Fourier transform of Ï_{H}\left(\mathfrak{z}\right)-x dense in an appropriate non-archimedean function space?â This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice \mathbb{Z}^{d} for any d\geq1.
Title: [2411.08206v2] Lua API and benchmark design using 3n+1 sequences: Comparing API elegance and raw speed in Redis and YottaDB databases
Authors: Berwyn Hoyt
Date: 2024-11-12
Abstract:
Elegance of a database API matters. Frequently, database APIs suit the database designer, rather than the programmerâs desire for elegance and efficiency. This article pursues both: firstly, by comparing the Lua APIs for two separate databases, Redis and YottaDB. Secondly, it looks under the API covers at how object orientation can help to retain API efficiency. Finally, it benchmarks both databases using each API to implement a 3n+1 sequence generator (of Collatz Conjecture fame). It covers the eccentricities of the Lua APIs, the databases, and the nifty choice of benchmark tool, presenting benchmark results of each databaseâs unique design.
Title: [2411.08084v2] Application of Operator Theory for the Collatz Conjecture
Authors: Takehiko Mori
Date: 2024-11-12
Abstract:
The Collatz map (or the 3n{+}1-map) f is defined on positive integers by setting f(n) equal to 3n+1 when n is odd and n/2 when n is even. The Collatz conjecture states that starting from any positive integer n, some iterate of f takes value 1. In this study, we discuss formulations of the Collatz conjecture by C^{*}-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the C^{*}-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated C^{*}-algebras.
Title: [2409.07929v1] General Dynamics and Generation Mapping for Collatz-type Sequences
Authors: Gaurav Goyal
Date: 2024-09-12
Abstract:
Let an odd integer (\mathcal{X}) be expressed as \left\{\sum\limits_{M > m}b_M2^M\right\}+2^m-1, where b_M\in\{0,1\} and 2^m-1 is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to 2^1-1. For the 3\mathcal{Z}+1 sequence, the Governor occurring in the Trivial cycle is 2^1-1, while for the 5\mathcal{Z}+1 sequence, the Trivial Governors are 2^2-1 and 2^1-1. Therefore, in these specific sequences, the Collatz function reduces the Governor 2^m - 1 to the Trivial Governor 2^{\mathcal{T}} - 1. Once this Trivial Governor is reached, it can evolve to a higher index Governor through interactions with other terms. This feature allows \mathcal{X} to reappear in a Collatz-type sequence, since 2^m - 1 = 2^{m - 1} + \cdots + 2^{\mathcal{T} + 1} + 2^{\mathcal{T}}+(2^{\mathcal{T}}-1). Thus, if \mathcal{X} reappears, at least one odd ancestor of \left\{\sum\limits_{M > m}b_M2^M\right\}+2^{m-1}+\cdots+2^{\mathcal{T}+1}+2^{\mathcal{T}}+(2^{\mathcal{T}}-1) must have the Governor 2^m-1. Ancestor mapping shows that all odd ancestors of \mathcal{X} have the Trivial Governor for the respective Collatz sequence. This implies that odd integers that repeat in the 3\mathcal{Z} + 1 sequence have the Governor 2^1 - 1, while those forming a repeating cycle in the 5\mathcal{Z} + 1 sequence have either 2^2 - 1 or 2^1 - 1 as the Governor. Successor mapping for the 3\mathcal{Z} + 1 sequence further indicates that there are no auxiliary cycles, as the Trivial Governor is always transformed into a different index Governor. Similarly, successor mapping for the 5\mathcal{Z} + 1 sequence reveals that the smallest odd integers forming an auxiliary cycle are smaller than 2^5. Finally, attempts to identify integers that diverge for the 3\mathcal{Z} + 1 sequence suggest that no such integers exist.
Title: [2408.02527v1] Rich dynamical behaviors from a digital reversal operation
Authors: Yannis Almirantis, Wentian Li
Date: 2024-08-05
Abstract:
An operation that maps one natural number to another can be considered as a dynamical system in \mathbb{N}^+. Some of such systems, e.g. the mapping in the so-called 3x+1 problem proposed by Collatz, is conjectured to have a single global attractor, whereas other systems, e.g. linear congruence, could be ergodic. Here we demonstrate that an operation that is based on digital reversal, has a spectrum of dynamical behaviors, including 2-cycle, 12-cycle, periodic attractors with other cycle lengths, and diverging limiting dynamics that escape to infinity. This dynamical system has infinite number of cyclic attractors, and may have unlimited number of cycle lengths. It also has potentially infinite number of diverging trajectories with a recurrent pattern repeating every 8 steps. Although the transient time before settling on a limiting dynamics is relatively short, we speculate that transient times may not have an upper bound.
Title: [2408.00805v1] On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations
Authors: T. Raptis
Date: 2024-07-29
Abstract:
The celebrated 3x+1 problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula f(x)+1 with a unique fixed point. The resultant formula f(x) is also found to coincide with that of the discrete derivative of the sorted sequence of fixed points of the reflection operator on even binary palindromes of fixed even length \textit{2k} in any interval [0\cdots2^{2k}-1]. A set of equivalent reformulations of the problem are also presented.
Title: [2406.08498v2] A matricial view of the Collatz conjecture
Authors: Pietro Paparella
Date: 2024-05-08
Abstract:
In this note, it is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the aperiodic Collatz conjecture.
Title: [2404.12279v2] Push-forward of geometric distributions under Collatz iteration: Part 1
Authors: Mary Rees
Date: 2024-04-18
Abstract:
Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under n Collatz iterates, modulo 2^p, is usefully close to uniform distribution on the integers modulo 2^p, if p/n is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both 0 and 1 for the coefficients in the dyadic expansions of -3^{-\ell } on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved.
Title: [2403.19699v1] Symbolic Dynamic Formulation for the Collatz Conjecture: I. Local and Quasi-global Behavior
Authors: Eric Sakk
Date: 2024-03-20
Abstract:
In this work, a symbolic dynamical formulation based upon discrete iterative mappings derived from the Collatz conjecture is introduced. It is demonstrated that this formulation naturally induces a ternary alphabet useful for characterizing the expansive and dissipative behavior of generated itineraries. Furthermore, local and quasi-global analyses indicate cyclic behaviors that should prove useful for describing global stability properties of itineraries. Additionally, techniques for generating arbitrarily long divergent itineraries are presented. Finally, this symbolic formulation allows itineraries to be grouped into sequence families that retain equivalent dynamical behavior.
Title: [2403.04777v1] Specifying and Verifying the Convergence Stairs of the Collatz Program
Authors: Ali Ebnenasir
Date: 2024-02-29
Abstract:
This paper presents an algorithmic method that, given a positive integer j, generates the j-th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly j applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the j-th convergence stair for any j > 0. The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. To the best of our knowledge, this is the first time that such a general specification is provided, which can have significant applications in analyzing and testing the behaviors of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.
Title: [2402.03276v3] An approximation of the Collatz map and a lower bound for the average total stopping time
Authors: Manuel Inselmann
Date: 2024-02-05
Abstract:
Define the map \mathsf{T} on the positive integers by \mathsf{T}(m)=\frac{m}{2} if m is even and by \mathsf{T}(m)=\frac{3m+1}{2} if m is odd. Results of Terras and Everett imply that, given any Δ>0, almost all m\in\mathbb{Z}^+ (in the sense of natural density) fulfill (\frac{\sqrt{3}}{2})^km^{1-Δ}\leq \mathsf{T}^k(m)\leq (\frac{\sqrt{3}}{2})^km^{1+Δ} simultaneously for all 0\leq k\leq α\log m with α=(\log 2)^{-1}\approx 1.443. We extend this result to α=2(\log\frac{4}{3})^{-1}\approx 6.952, which is the maximally possible value. Set \mathsf{T}_{\min}(m):=\min_{n\in\mathbb{N}}\mathsf{T}^n(m). As an immediate consequence, one has \mathsf{T}_{\min}(m)\leq\mathsf{T}^{\left\lfloor2(\log\frac{4}{3})^{-1}\log m\right\rfloor}(m)\leq m^Δ for almost all m\in\mathbb{Z}^+ for any given Δ>0. Previously, Korec has shown that \mathsf{T}_{\min}(m)\leq m^Δ for almost all m\in\mathbb{Z}^+ if Δ>\frac{\log3}{\log4}, and recently Tao proved that \mathsf{T}_{\min}(m)\leq f(m) for almost all m\in\mathbb{Z}^+ (in the sense of logarithmic density) for all functions f diverging to \infty. Denote by Ï(m) the minimal n\in\mathbb{N} for which \mathsf{T}^n(m)=1 if there exists such an n and set Ï(m)=\infty otherwise. As another application, we show that \liminf_{x\rightarrow\infty}\frac{1}{x\log x}\sum_{m=1}^{\lfloor x\rfloor}Ï(m)\geq 2(\log\frac{4}{3})^{-1}, partially answering a question of Crandall and Shanks. Under the assumption that the Collatz Conjecture is true in the strong sense that Ï(m) is in O(\log m), we show that \lim_{x\rightarrow\infty}\frac{1}{x\log x}\sum_{m=1}^{\lfloor x\rfloor}Ï(m)= 2(\log\frac{4}{3})^{-1}.
Title: [2402.00001v3] Using binary string to prove the Collatz conjecture
Authors: Jishe Feng
Date: 2023-09-20
Abstract:
We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our analysis focuses on examining the iteration sequence (or composition) of the Collatz function and its reduced variant, which serves as an analog to the inverse function, to scrutinize the validity of the Collatz conjecture. To substantiate this conjecture, we incorporate binary strings into an algebraic formula that captures the essence of the Collatz sequence. By this means, we transform discrete powers of 2 into continuous counterparts, ultimately culminating in the smallest natural number, 1. Consequently, the sequence generated through infinite iterations of the Collatz function emerges as an eventually periodic sequence, thereby validating an enduring 87-year-old conjecture.
Title: [2401.17241v1] Almost all orbits of an analogue of the Collatz map on the reals attain bounded values
Authors: Manuel Inselmann
Date: 2024-01-30
Abstract:
Motivated by a balanced ternary representation of the Collatz map we define the map C_\mathbb{R} on the positive real numbers by setting C_\mathbb{R}(x)=\frac{1}{2}x if [x] is even and C_\mathbb{R}(x)=\frac{3}{2}x if [x] is odd, where [x] is defined by [x]\in\mathbb{Z} and x-[x]\in(-\frac{1}{2},\frac{1}{2}]. We show that there exists a constant K>0 such that the set of x fulfilling \liminf_{n\in\mathbb{N}}C_\mathbb{R}^n(x)\leq K is Lebesgue-co-null. We also show that for any Δ>0 the set of x for which (\frac{3^{\frac{1}{2}}}{2})^kx^{1-Δ}\leq C_\mathbb{R}^k(x)\leq (\frac{3^{\frac{1}{2}}}{2})^kx^{1+Δ} for all 0\leq k\leq \frac{1}{1-\frac{\log_23}{2}}\log_2x is large for a suitable notion of largeness.
Title: [2401.12781v2] On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$
Authors: Manuel Inselmann
Date: 2024-01-23
Abstract:
Define the map T_1 on \mathbb{F}_2[x] by T_1(f)=\frac{f}{x} if f(0)=0 and T_1(f)=\frac{(x+1)f+1}{x} if f(0)=1. For a non-zero polynomial f let Ï_1(f) denote the least natural k number for which T_1^{k}(f)=1. Define the average stopping time to be Ï_1(n)=\frac{\sum_{f\in \mathbb{F}_2[x], \text{deg}(f)=n }Ï_1(f)}{2^n}. We show that \lim_{n\rightarrow\infty}\frac{Ï_1(n)}{n}=2, confirming a conjecture of Alon, Behajaina, and Paran. Furthermore, we give a new proof that Ï_1(f)\in O(\text{deg}(f)^{1.5}) for all f\in\mathbb{F}_2[x]\setminus\{0\}.
Title: [2401.03210v1] On the stopping time of the Collatz map in $\mathbb{F}_2[x]$
Authors: Gil Alon, Angelot Behajaina, Elad Paran
Date: 2024-01-06
Abstract:
We study the stopping time of the Collatz map for a polynomial f \in \mathbb{F}_2[x], and bound it by O({\rm deg} (f)^{1.5}), improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.
Title: [2312.17043v4] Collatz-Weyl Generators: High Quality and High Throughput Parameterized Pseudorandom Number Generators
Authors: Tomasz R. DziaĆa
Date: 2023-12-28
Abstract:
We introduce the Collatz-Weyl Generators, a family of uniform pseudorandom number generators (PRNGs) which are based on generalized Collatz mappings, derived from the Collatz conjecture and Weyl sequences. The high-quality statistical properties of our generators is demonstrated by the fact that they pass stringent randomness tests used by the research and standardization community. The proposed Collatz-Weyl Generators have a number of important properties, including solid mathematical foundations, enablement of high throughput and low latency implementation, small code and/or ASIC size, enablement of producing multiple independent streams and potential of support of cryptographic applications.
Title: [2312.00390v2] The Collatz map analogue in polynomial rings and in completions
Authors: Angelot Behajaina, Elad Paran
Date: 2023-12-01
Abstract:
We study an analogue of the Collatz map in the polynomial ring R[x], where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in R[x] is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on \mathbb{F}_p[x] and \mathbb{F}_2[x], respectively. We also consider the Collatz map on the ring of formal power series R[[x]] when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring \mathbb{Z}_2 of 2-adic integers, extending previous results of Lagarias.
Title: [2311.16765v3] Observations towards a proof of the Collatz conjecture using $2^{j}k+x$ number series
Authors: J. Stöckl
Date: 2023-11-28
Abstract:
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller number, all full sequences lead to 1 with a finite stopping time, the problem could be reduced to more structured shorter sequences. It is shown that this sequences exist and follow consistent rules. Potential features of an infinite cycle, also leading to a smaller number, are also discussed. Further, an argument for only one possible closed cycle is given for the special sequence of alternating odd and even steps as well as arguments that infinite cycles must exist. Using the observation that periodic behavior is exists an additional argument is provided the probability of subsets which will end up at a number smaller the initial value possibly even of N will indeed end up at unity in the Collatz problem [1]. The work is to be seen as in progress and shared as an contribution to discussion rather than a concrete publication.
Title: [2311.12038v1] Modular Arithmetic Study on Ascending and Descending Operations Related with Collatz Conjecture
Authors: Kyo Jin Ihn
Date: 2023-11-17
Abstract:
Sequence of numbers generated by the recurrence relation based on the Collatz conjecture is investigated. An arithmetic operation on the Collatz conjecture is called descending operation, and ascending operation is carried out reversely to the descending operation. Study on the 3x+1 problem against every integer is relevant to that of descending operation against every odd. Any odds can be generated by ascending operations against odds not multiple of 3, and vise versa. Ascending operations against an odd not a multiple of 3 can generate infinite number of odds in the next generation. The odds multiple of 3 are terminal numbers of sequences that no further ascending operations are possible. Descending operations against the odds 1,5,21,.. reach 1, and otherwise reach the odds greater than or equal to 5. No two odds except 1 in each sequence are the same, and the each descending sequence is unique. Every descending sequence can start from terminal numbers, and reaches ultimately the origin of 1 as indicated by the Collatz conjecture. Sequences of odds in the same pattern ignoring the sizes of the odds are generated by ascending operations with the same number of doubling operations against the odds in modular arithmetic.
Title: [2310.13930v1] Note on Collatz conjecture
Authors: Abdelrahman Ramzy
Date: 2023-10-21
Abstract:
In this paper, we show that if the numbers in the range [1,2^n] satisfy Collatz conjecture, then almost all integers in the range [2^n+1,2^{n+1}] will satisfy the conjecture as n \to \infty. The previous statement is equivalent to claiming that almost all integers in [2^n+1,2^{n+1}] will iterate to a number less than 2^n. This actually has been proved by many previous results. But in this paper we prove this claim using different methods. We also utilize our assumption on numbers in [1,2^n] to show that there are a set of integers (denoted by p\operatorname{-}C) whose seem to be not iterating to a number less than 2^n, but since they are connected to Collatz numbers in [1,2^n] they eventually will iterate to 1. We address the distribution of p\operatorname{-}C and give an explicit formula which computes a lower bound to the number of these integers. We also show (computationally) that the number of p\operatorname{-}C in a given interval is proportional to the numbers of another set of incidental Collatz numbers in the same interval (whose distribution is completely unpredictable).
Title: [2310.13035v2] Collatz conjecture becomes theorem
Authors: GraĆŒyna Mirkowska, Andrzej Salwicki
Date: 2023-10-19
Abstract:
The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatzâs theorem cannot be proved in any elementary number theory completes the main result.
Title: [2310.08749v1] On the Convergence of the Density of Terras' Set
Authors: Idris Assani, Ethan Ebbighausen
Date: 2023-10-12
Abstract:
The Collatz Conjectureâs connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of convergence. This rate gives a preliminary result on the Triangle Conjecture, which describes a set nodes that would dominate L^{C} = \{y \in \N \, | \, T^{k}(y) > y , \,\forall k \in \N \}. Second, we extend prior arguments to show that the construction of several classes of measures imply the bounded trajectories piece of the Collatz Conjecture.
Title: [2309.09991v3] The Proof of the Collatz Conjecture
Authors: Chin-Long Wey
Date: 2023-09-15
Abstract:
The 3n+1, or Collatz problem, is one of the hardest math problems, yet still unsolved. The Collatz conjecture is to prove or disprove that the Collatz sequences COL(n) always eventually reach the number of 1, for all n belongs to N+ (all positive integers). The Syracuse conjecture is a (2N+1)-version of Collatz conjecture, where (2N+1) is all odd integers. The Syracuse and Collatz problem can be conceptually described by a tree trunk and branches. The trunk is made of the junctions that produce the main branches, where J0=1 is the root junction. Each branch consists of active and dead junctions, where only the active junctions are capable of producing new sub-branches. Conceptually assuming the trunk and branches can grow indefinitely and can also absorb nutrients from the root. As the tree grows indefinitely, all N+ (2N+1) are included for the Collatz (Syracuse) sequence. This paper develops both inverse Collatz (Syracuse) functions to construct the tree trunk and branches starting from the root junction J0=1 and assign the also positive (odd) integers to all junctions. To verify the Collatz (Syracuse) sequences always eventually reach the number of 1, this paper also develops the PathFinding algorithm. Given n belongs to N+ (2N+1), the algorithm finds a path from n to the root junction J0=1 by the virtual tree structure to prove both Syracuse and Collatz conjectures.
Title: [2309.04807v1] Linear convergence of the Collatz method for computing the Perron eigenpair of primitive dual number matrix
Authors: Yongjun Chen, Liping Zhang
Date: 2023-09-09
Abstract:
Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method was also extended to find Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix tends to zero if and only if the spectral radius of its standard part less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.
Title: [2308.01214v2] An optimal convergent Collatz algorithm
Authors: Juan Carlos Riano-Rojas
Date: 2023-08-02
Abstract:
In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated that the proposed theory is correct with several examples; as a sector of mathematicians believes that the Collatz conjecture fails in some power of 3. The algorithm was applied to the first 600 powers of 3, where the convergence of the proposed algorithm was verified.
Title: [2308.01181v2] On Collatz Conjecture for binary polynomials
Authors: Luis H. Gallardo, Olivier Rahavandrainy
Date: 2023-08-02
Abstract:
We build a variant of Collatz Conjecture for polynomials over \mathbb{F}_2 and we prove that it is solved. By the way, we give several examples.
Title: [2307.11990v5] On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem
Authors: Yagub N. Aliyev
Date: 2023-07-22
Abstract:
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear combinations are integers. The discussed results are demonstrated on some examples. In some particular cases the obtained results can be used to explain some patterns of digits in p-adic representations of the terms of the rational cycles.
Title: [2306.16140v2] Dual Number Matrices with Primitive and Irreducible Nonnegative Standard Parts
Authors: Liqun Qi, Chunfeng Cui
Date: 2023-06-28
Abstract:
In this paper, we extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. One motivation of our research is to consider probabilities as well as perturbation, or error bounds, or variances, in the Markov chain process. We show that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector. The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix. We present an explicit formula to compute the dual part of this positive dual number eigenvalue. The Collatz minimax theorem also holds here.The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all. An algorithm based upon the Collatz minimax theorem is constructed. The convergence of the algorithm is studied. We show the upper bounds on the distance of stationary states between the dual Markov chain and the perturbed Markov chain. Numerical results on both synthetic examples and dual Markov chain including some real world examples are reported.
Title: [2306.15284v2] Are the Collatz and abc conjectures related?
Authors: Olivier Rozier
Date: 2023-06-27
Abstract:
The Collatz and abc conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the abc conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcutââ form. To obtain sharper bounds in this context, we are led to consider a small subset of the abc-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.
Title: [2306.14635v2] The Collatz problem (a*q+-1,a=1,3,5,...) from the point of view of transformations of Jacobsthal numbers
Authors: Petro Kosobutskyy
Date: 2023-06-26
Abstract:
In the paper, from the point of view of recurrent numbers of the Jacobsthal type, the Collatz problem with the general aq±1 function of conjecture odd positive integers q from the set of natural numbers is investigated. Formulated branching rules from nodes with generalized Jacobsthal numbers of the so-called Jacobsthal tree. It is shown that Collatz trajectories are formed in the reverse direction of the Jacobsthal tree. It is shown that unlike the classical Collatz problem, in which the Collatz sequence for a finite number of iterations leads to the unit element, for functions with factors 3 and 5, the Collatz sequence for a finite number of iterations does not reach the unit element. An analytical model is proposed that relates the iteration number of doubling (or halving the number in the reverse direction) number between two odd-numbered branch nodes. It is shown that for an arbitrary number, Collatz and Jacobsthal trajectories develop in the respective directions, and the Jacobsthal trajectory for a finite number of iterations leads to an element that is a multiple of the factor a in the function aq±1.
Title: [2306.10333v2] Bounded fixed point sets and Krasnoselskii iterates of Thompson metric nonexpansive maps
Authors: Brian Lins
Date: 2023-06-17
Abstract:
We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompsonâs metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps converge to a fixed point when one exists. For maps that are also order-preserving, we give simple necessary and sufficient conditions in terms of upper and lower Collatz-Wielandt numbers for the fixed point set to be nonempty and bounded in Thompsonâs metric. When the map is also real analytic, these conditions are both necessary and sufficient for the map to have a unique fixed point and for all iterates of the map to converge to the fixed point. We demonstrate how these results apply to certain nonlinear matrix equations on the cone of positive definite Hermitian matrices.
Title: [2305.10117v1] A new conjecture equivalent to Collatz conjecture
Authors: Giulio Masetti
Date: 2023-05-17
Abstract:
In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz conjecture.