相关论文: The Collatz 3n+1 Conjecture is Unprovable
The Collatz conjecture is a famous math problem that was introduced by Lothar Collatz in 1937, and nobody has yet succeeded in proving or disproving it. In this article, I will analyze this problem with a new approach and I will discuss my…
The Collatz function is defined as C(n) = n / 2 if n is even and C(n) = 3n + 1 if n is odd. The Collatz conjecture states that every sequence generated by the Collatz function ends with the cycle (4, 2, 1) after a finite number of…
This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this…
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof…
In this paper, we discuss the well known 3x+1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and an invertible map that are intrinsically related to…
The scope of the present work is to explain why it is true that all N have a distinct position in The Collatz Tree (The Collatz Graph)
Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer…
In this article, we give two different proofs of why the Collatz Conjecture is false.
In this article, we define a very important sequence of functions, all the functions of this sequence present behaviors very close to that of the Collatz function. The study of such functions allows us to obtain very interesting results…
This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation…
The infamous 3x+1 conjecture spread by Lothar Collatz in 1952, despite its elementary formulation, remained unproved for over 60 years. From the heuristical probabilistic approach to the complex mapping of the algorithm, the scientific…
Let $g$ be a map from the set of positive integers into itself defined as follows: Let $x$ be a positive integer. If $x$ is odd, then $g(x)=3x+1$, and if $x$ is even, then $g(x)=x/2$. The $3x+1$ conjecture, also called the Collatz…
The Collatz problem is generalized into $3n + 3^k$ problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the $3n + 3^k$ sequence converges to the cycle passing through the…
We give a generalization of Collatz conjecture or 3n+1 problem on 2-adic completion of Q. A isometric of $Q_2$ provides information on the average behavior of the firsts terms of the sequence according to the class of $u_0$ modulo $2^m$. A…
The $3x+1$ problem, also called the Collatz conjecture, is a very interesting unsolved mathematical problem related to computer science. This paper generalized this problem by relaxing the constraints, i.e., generalizing this deterministic…
We represent the generalized Collatz function with the recursive ruler function r(2n) = r(n) + 1 and r(2n + 1) = 1. We generate even-only and odd-only Collatz subsequences that contain significantly fewer elements term by term, to 2 and 1,…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
In this article, we reduce the unsolved problem of convergence of Collatz sequences to convergence of Collatz sequences of odd numbers that are divisible by 3. We give an elementary proof of the fact that a Collatz sequence does not…
We describe a new algorithm for verifying the Collatz conjecture for all n < 2^N for some fixed N. The algorithm takes less than twice as long to verify convergence for all n < 2^{N+1} as it does to verify convergence for all n < 2^N. We…
The Collatz conjecture implies that an iterated function sequence under a certain linear operator, beginning with a certain complex valued function, must converge to a certain complex function.