Collatz Conjecture Research Articles

Collatz Conjecture (3N+1) Solution

Collatz Conjecture (3x+1) or in some literature as 3N+1 is a problem because it works in the way that if you take any positive number, if it is an odd number you multiply it by three (3) then add one (1). On the other hand, if it is an even number, you divide it by two (2). Eventually, all positive numbers decrease to one (1). One (1) is odd, so multiply it by three (3) is three (3) and add one (1) is four (4). Four (4) is even, so divide it by two (2) is two (2). Two (2) is also even, so divide it by two (2) is one (1) again. All positive numbers end up in the loop (4-2-1). This loop is like a numerical lock. Therefore, the solution of this problem will have to be a numerical key results to all positive numbers.

Analysis of Collatz Conjecture Rules

Abstract A proof of the Collatz Conjecture is presented. Changing the perspective of the problem from looking at the pattern of the positive integers to looking at the conjecture rules made the proof possible. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and values continually increasing to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer. JEL classification numbers: CO2; C65. Keywords: Collatz Conjecture, Proof, Rules, Dendritic pathway.

A matricial view of the Collatz conjecture

It is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the Collatz conjecture. Our main result extends the previous work of Alves et al. and clarifies a conjecture made by Cardon and Tuckfield.

Functional Analysis Approach to the Collatz Conjecture

Functional Analysis Approach to the Collatz Conjecture

Some extensions of Collatz (periodic) conjecture

The Collatz conjecture is a famous conjecture aiming to estimate the behaviors of positive integers after a series of iterations of the Collatz function f(n), where f(n)={3n+12 ifnis odd,n2 ifnis even. Collatz conjecture has attracted a great deal of attention since it was put forward in 1930s due to the surprising nature. However, after strenuous efforts, many mathematicians think the current knowledge seems insufficient to solve Collatz conjecture, although it appears in a very simple form. As a replacement, a “periodic” (weaker) version of Collatz conjecture (called Collatz periodic conjecture) was proposed and discussed. In this paper, we would like to explore what properties Collatz (periodic) conjecture could imply. Maybe it is another possible direction to confirm or disprove Collatz (periodic) conjecture.

Collatz Conjecture Solution

The Collatz conjecture, also known as the 3n + 1 conjecture, stands as one of the most enduring unsolved problems in mathematics. Proposed by German mathematician Lothar Collatz in 1937, the conjecture poses a deceptively simple question: can iteratively applying two basic arithmetic operations—dividing even numbers by 2 and multiplying odd numbers by 3 and adding 1—lead any positive integer to converge to the value 1? Despite extensive computational verification for vast ranges of integers, a proof or disproof remains elusive. This abstract explores the historical background, various attempts at proving the conjecture, and its implications across mathematical domains.

A Proof of Collatz Conjecture Using Pyramid Fractions

In this paper, I introduce a new concept of representing numbers in base; in other words, I have found new series for any number similar to the series that could be written according to the collatz sequence which is called zi(n) in this article. These series need to end with 1. Then, we use two sets of rules to make a diagram which then proves the existence of such series for any number

On the Verification and the Estimation for Two Expansion Problems of L. Collatz Conjecture

[9] presented two expansions 5n+1 problem and 7n+1 problem for L. Collatz Conjecture which is an unsolved problem, and confirmed two expansions are correct for all n<100000 (=10^5). And [10] proposed two approximate formulas of the average number of steps for 5n+1 problem and 7n+1 problem respectively. In this paper, we confirm that two expansions are correct for all n<1000000000000 (=10^12), and propose two high precision formulas of the average number of steps for 5n+1 problem and 7n+1 problem respectively. Keywords: Collatz Conjecture, 5n+1 problem, 7n+1 problem, Verification, average number of steps.

Feedback resonance in Fermi–Pasta–Ulam chain

A controlled version of the celebrated Fermi–Pasta–Ulam problem is introduced. The feedback control algorithm based on Speed-gradient approach is proposed and analyzed by extensive computer simulation. It is demonstrated that the feedback controlled system tends to approximate equipartition state much faster than it happens in the open loop (classical) system. Apparently a substantial change in system behavior is achieved with a small intensity control: less than 0.5% of the total system energy. Such a behavior has been observed and analyzed previously for oscillators with one degree-of-freedom (DOF) under the name of feedback resonance (Fradkov A.L., Physica D, 128(1999), 159-168. The results of this paper suggest that feedback resonance phenomenon may be observed in multi-DOF oscillator chains, particularly in FPU chain.

Conjectures analogous to the Collatz conjecture

In this paper we introduce some conjectures analogous to the well-known Collatz conjecture.

Linear dynamics of an operator associated to the Collatz map

In this paper, we study the dynamics of an operator T \mathcal T naturally associated to the so-called Collatz map, which maps an integer n ≥ 0 n \geq 0 to n / 2 n / 2 if n n is even and 3 n + 1 3n + 1 if n n is odd. This operator T \mathcal T is defined on certain weighted Bergman spaces B ω 2 \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that T \mathcal T is hypercyclic on B ω 2 \mathcal B ^2 _\omega , independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight ω \omega , we show that T \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic.

Neka razmatranja o ukupnom vremenu zaustavljanja za Kolacov problem

Introduction/purpose: The Collatz conjecture has been considered and the stopping time needed for the recursive transformation to end has been investigated. Methods: A statistical analysis on the stopping time has been used. Results: The statistical approach shows that the probability of finding an infinite stopping time, that is a violation of the Collatz conjecture, is extremely low. Conclusion: Picking precisely one particular atom in the Universe is still more favorable, by more than 61 orders of magnitude, than encountering an infinite total stopping time.

The Definitive Proof that Collatz Conjecture is True

The Definitive Proof that Collatz Conjecture is True

Generating Powerful Encryption Keys for Image Cryptography With Chaotic Maps by Incorporating Collatz Conjecture

Generating Powerful Encryption Keys for Image Cryptography With Chaotic Maps by Incorporating Collatz Conjecture

Lie groups in the symmetric group: Reducing Ulam's problem to the simple case

Ulam asked whether all Lie groups can be represented faithfully on a countable set. We establish a reduction of Ulam's problem to the case of simple Lie groups.In particular, we solve the problem for all solvable Lie groups and more generally Lie groups with a linear Levi component. It follows that every amenable locally compact second countable group acts faithfully on a countable set.

The Proof of the Collatz Conjecture

In this paper, I intend to prove the Collatz conjecture. In the following, I will prove that theCollatz conjecture is true for any positive integer and also for natural numbers. That is, aftersufficiently using the rules of the Collatz conjecture, we must reach 1. In fact, I aim to give adefinite answer to the question that has not been answered since 1937: Does the Collatz sequenceeventually reach 1 for all-natural or positive integer initial values? My answer is yes.

Half an Induction Proof of the Collatz Conjecture

Summary In a straightforward proof by induction of the Collatz conjecture, it is easy to show that the statement is true for n = 1 (the base case) but proving the induction step is more challenging—in fact, no one has done so to date. In this work, a different induction proof of the conjecture, based on a binary representation of the starting integer, is given in which it is possible to prove the induction step but there is no known proof for the base case. In fact, if one could prove the base case, then the Collatz conjecture is true.

Proof of the Collatz Conjecture

Abstract The Collatz conjecture (or 3n+1 problem) has been explored for about 86 years. In this article, we prove the Collatz conjecture. We will show that this conjecture holds for all positive integers by applying the Collatz inverse operation to the numbers that satisfy the rules of the Collatz conjecture. Finally, we will prove that there are no positive integers that do not satisfy this conjecture. Mathematics Subject Classification (2020): 11A25, 11A51, 11B50, 11B75. Keywords: Collatz operation, Collatz inverse operation, Collatz numbers.

From Collatz Conjecture to chaos and hash function

The non-linear property of Chaos is a promising approach to information security, and many accomplishments have been made by combining Chaos with several sub-security domains, including chaos-based stream ciphers, block ciphers, image encryption, and hash functions. Most Chaos-based hash functions are insecure or inefficient due to their dependence on complex, attacked multi-dimensional maps or uncertain, weak one-dimensional maps like logistic and tent. The Collatz Conjecture is a mystery that has stumped mathematicians for decades and still has not been solved. This paper aims to introduce a novel approach to addressing current security challenges by utilizing our generalized Collatz process to create a chaos-based hash function. By leveraging the unpredictable behaviour of the Collatz sequence, the proposed hash function aims to enhance ergodicity and entropy properties, thereby making it well-suited for cryptographic applications. In the proposed method, the chaotic variables are governed by cryptographic keys, crucial in generating data sequences. These sequences are then utilized within the diffusion and confusion structures of the hashing function. The design of the chaos-hash model is carefully optimized to exhibit desirable characteristics such as randomness, collision resistance, uniformity, sensitivity to initial conditions, speed, and resistance against cryptanalysis.The primary goal of this research is to develop a robust and efficient chaos-based hash function that addresses the requirements of cryptographic systems. By incorporating the Collatz process and carefully considering key-controlled variables, the proposed model aims to offer enhanced security properties while meeting the necessary criteria for a reliable and effective hashing mechanism. The effectiveness and dependability of the proposed hash function are evaluated by comparing it with two well-known hash algorithms, namely SHA-3 and SHA-2, as well as several other Chaos-based hash algorithms. Through comprehensive evaluations conducted under various circumstances and with different datasets, the suggested hash function consistently outperforms the state-of-the-art alternatives. Experiments involving diverse input scenarios consistently demonstrate that the proposed hash function surpasses the performance of the current leading alternatives. Its superior performance is observed across various evaluation metrics, confirming its effectiveness in generating reliable and secure hash values. The results obtained from the comparative analysis highlight the superiority of the proposed hash function over existing alternatives, validating its potential as a robust solution for various cryptographic applications.

A multithreaded CUDA and OpenMP based power‐aware programming framework for multi‐node GPU systems

SummaryIn the article, we have proposed a framework that allows programming a parallel application for a multi‐node system, with one or more graphical processing units (GPUs) per node, using an OpenMP+extended CUDA API. OpenMP is used for launching threads responsible for management of particular GPUs and extended CUDA calls allow to transfer data and launch kernels on local and remote GPUs. The framework hides inter‐node MPI communication from the programmer. For optimization, the implementation takes advantage of the MPI_THREAD_MULTIPLE mode allowing: multiple threads handling distinct GPUs as well as overlapping communication and computations transparently using multiple CUDA streams. The solution allows data parallelization across available GPUs in order to minimize execution time and supports a power‐aware mode in which GPUs are automatically selected for computations using a greedy approach in order not to exceed an imposed power limit. We have implemented and benchmarked three parallel applications including: finding the largest divisors; verification of the Collatz conjecture; finding patterns in vectors. These were tested on three various systems: a GPU cluster with 16 nodes, each with NVIDIA GTX 1060 GPU; a powerful 2‐node system—one node with 8 NVIDIA Quadro RTX 6000 GPUs, the second with 4 NVIDIA Quadro RTX 5000 GPUs; a heterogeneous environment with one node with 2 NVIDIA RTX 2080 and 2 nodes with NVIDIA GTX 1060 GPUs. We demonstrated effectiveness of the framework through execution times versus power caps within ranges of 100–1400 W, 250–3000 W, and 125–600 W for these systems respectively as well as gains from using two versus one CUDA streams per GPU. Finally, we have shown that for the testbed applications the solution allows to obtain high speed‐ups between 89.3% and 97.4% of the theoretically assessed ideal ones, for 16 nodes and 2 CUDA streams, demonstrating very good parallel efficiency.