Collatz Conjecture Research Articles

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

On the Collatz Conjecture

The Collatz conjecture (or Syracuse conjecture) states: all Syracuse sequences converge to 1. We present a Syracuse sequence, and we prove that the conjecture is true, first by using the fact that all convergent integer sequences are eventually constant. We then prove wrong 2 hypotheses: the case where the sequence tends to infinity, and the case where the sequence has no limit and is eventually periodic. We conclude by elimination, afterward.

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Extended dynamic mode decomposition for two paradigms of non-linear dynamical systems

We apply the Koopman operator theory and Extended Dynamic Mode Decomposition in two non-linear dynamical systems. The first one is the 3x+1-system which follows by the so-called Collatz conjecture and it is discrete. The second one is the SIR-model which is continuous and is used to describe the evolution of a disease through some population. Koopman operator captures the dynamics of a non-linear system, however it is infinite dimensional. In this study, we approximate the aforementioned systems with finite dimensional linear systems which are defined on some augmented state space. In the case of the Collatz system, this approach is expected to give some more insight to the 3x+1-problem. In the case of SIR-model, using some suitable dictionary of observables, we obtain a linear system which approximates the trajectories of the system and can be used for prediction and control purposes.

Explicit exactly energy-conserving methods for Hamiltonian systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Secure Repackage-Proofing Framework for Android Apps Using Collatz Conjecture

App repackaging has been raising serious concerns about the health of the Android ecosystem, and repackage-proofing is an important mitigation against threat of such attacks. However, existing app repackage-proofing schemes were only evaluated against trivial adversaries simulated using analyzers for other purposes (e.g., disclosing privacy leakage vulnerabilities), hence were shown “effective” mainly because their key programming features were not even supported by those toolkits. Furthermore, existing works have also neglected dynamic adversaries capable of manipulating victim apps at runtime, making them vulnerable against such stronger opponents. In this article, we propose a novel repackage-proofing framework, which deploys distributed detection and response sites into the subject app's native partition to cross-verify all its code files. The detection sites transmit obtained integrity metrics to response sites via secure communication channels built on the subject app's own control flows using a specialized obfuscation technique based on Collatz conjecture, turning the repackage-proofing process into complicated implicit flows that are intrinsically difficult to be resolved due to the conjecture's nonlinear dynamical behaviors. We evaluated our framework using sophisticated Android data-flow analyzers. Results showed that our prototype effectively impeded analyses aiming to trace the information flows of its cross-verification.

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Proof of Collatz Conjecture: The Traps, a Flowchart, Their IDs (Number Identification), a Table or Summary, Collatz Conjecture Inequality and Analysis

Proof of Collatz Conjecture: The Traps, a Flowchart, Their IDs (Number Identification), a Table or Summary, Collatz Conjecture Inequality and Analysis

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Various Results on the Collatz Conjecture

Solution to the Collatz Conjecture

Solution to the Collatz Conjecture

Analyzing the Collatz Conjecture Using the Mathematical Complete Induction Method

In this paper, we demonstrate the Collatz conjecture using the mathematical complete induction method. We show that this conjecture is satisfied for the first values of natural numbers, and in analyzing the sequence generated by odd numbers, we can deduce a formula for the general term of the Collatz sequence for any odd natural number n after several iterations. This formula is used in one case that we analyze using the mathematical complete induction method in the process of demonstrating the conjecture.

An Estimation of Hailstone Radii using the Collatz Conjecture

There is an alleged connection between a much researched, yet unsolved math problem, the Collatz conjecture, and a quite common, damaging meteorological phenomenon: hailstorm. The relationship would be that, in sequences generated by Collatz algorithm, the way in which numbers rise and fall resembles hailstones going up and down inside a cloud, whence the name “Hailstone sequences”. The aim of this paper is two-fold: first, to use JavaScript to research on the Collatz conjecture with the perspective of a high-school student. Our algorithm tested a generalized form of the conjecture for multiple primes (3, 5, 7) and signs (+,). The Pearson correlation coefficient found between the initial value and, respectively, the total stopping time or the maximum value reached excluded any linear correlation. The second (and main) goal was to assess the hypothesis whether hailstones could indeed follow a Collatz-like function trajectory, studying the implication on the radii of them. Introducing the concept of conversion formula, we estimated the final radii for different functions (straight line, square-root, square, logarithmic, exponential), unit of measures (from Km to mm), and starting heights ranging from 4000m to 10000m, should the motion of hailstones behave like a Collatz function. In all but one case, we did not get radii believable in size, and reasonably randomly distributed. For the linear formula (in cm), the -test values between our estimated values and Nelson’s model values are above the critical values. Hence, we should reject the initial hypothesis.