Query: Collatz conjecture prove

title: A new method to prove the Collatz conjecture
abstract: The Collatz conjecture is an unsolved problem in mathematics which introduced by Lothar Collatz in 1937. Although the prize for the proof of this problem is 1 million dollar, nobody has succeeded in proving this conjecture. However in this article, we will discuss the results of the author's research and come closer to the proof of this conjecture.

title: Convergence of Collatz Sequences: Procedure to Prove the Collatz Conjecture
abstract: As Collatz conjecture is still to be proved, a method to arrive at the complete proof is explored here. Conceptually, the process relies on the pre proven sequence data and the method follows the confirmation of the convergence of the Collatz sequence for all the natural numbers in a sequential forward manner (ascending order of n) Hence, all the numbers less than n have been proved to converge before beginning to prove the convergence of the sequence for the number n. Initially, the the nature of problem was explored and explained the reason for some of the remarkable properties. Then, the pattern analysis of a first few sequences data led to the concept of pre proven sequence. Then, the origin of quartets and subsequently, the octet patterns are described. The "Even Odd" combinations and the concept of (8*v c) gave a clear picture for the overall process of evaluation for the convergence. Then onwards, individual set of numbers are analyzed and the convergence confirmations are arrived at. The process of computing involves the concept of combing. Finally, the compilation of all the sets together would give the completion of the proof. Here, the procedure for the computation is presented with example and the current status of computation >99% completed) while the laborious computation is in progress (for the remaining 1% It is expected that the length of the PSO number list is finite and all the numbers would converge.

title: Pruning the binary tree, proving the Collatz conjecture
abstract: The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n+1, connects all natural numbers to a single tree with 1 as its root. Pruning ever more numbers from this tree until the density of pruned numbers relative to the natural numbers is arbitrarily close to 100% would prove that all numbers belonged to the tree. Preliminarily, numbers divisible by 2 or 3 are pruned. A binary tree is reconnected that forks at each number towards an `upward' and a `rightward' number. Next, intermediate rightward numbers, upward numbers with a rightward parent, with a rightward grandparent, etcetera, are pruned, which accrues the density of pruned numbers arbitrarily close to 100% This proves the Collatz conjecture.

title: On the orbits associated with the Collatz conjecture
abstract: This article leans on previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. Unfortunately, they could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.

title: Classify Positive Integers to Prove Collatz Conjecture by the Mathematical Induction
abstract: Positive integers which are able to be operated to 1 by set operational rule of the Collatz conjecture and positive integers got via operations by the operational rule versus the set operational rule are one to one the same, thus we refer to converse operational routes, apply the mathematical induction, next classify positive integers to prove the Collatz conjecture by substeps according to beforehand prepared two theorems concerned.

title: A New Approach on Proving Collatz Conjecture
abstract: Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3*x+1 computation (when x is odd) and x/2 computation (when x is even) In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC) We propose Reduced Collatz Conjecture (RCC) any natural number x will return to an integer that is less than x. We prove that RCC is equivalent to CC. For proving RCC, we propose exploring laws of Reduced Collatz Dynamics (RCD) i.e. from a starting integer to the first integer less than the starting integer. RCC can also be stated as follows: RCD of any natural number exists. We prove that RCD is the components of original Collatz dynamics (from a starting integer to 1) i.e. RCD is more primitive and presents better properties. We prove that RCD presents unified structure in terms of (3*x+1)/2 and x/2, because 3*x+1 is always followed by x/2. The number of forthcoming (3*x+1)/2 computations can be determined directly by inputting x. We propose an induction method for proving RCC. We also discover that some starting integers present RCD with short lengths no more than 7. Hence, partial natural numbers are proved to guarantee RCC in this paper, e.g. 0 module 2; 1 module 4; 3 module 16; 11 or 23 module 32; 7, 15, or 59 module 128. The future work for proving CC can follow this direction, to prove that RCD of left portion of natural numbers exists.

title: Proving the Collatz Conjecture
abstract: Collatz sequences are formed by dividing an even number by two until it is odd. Then multiply by three and add one to get an even number. The Collatz conjecture states that if this process is repeated you always get back to one. Using geometric series summations we prove that a connected Collatz Structure exists, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

title: Proof of the Collatz Conjecture
abstract: Abdelkrim TAHA Abstract In this paper we prove the Collatz conjecture also known the 3x+1 problem. The conjecture let a0 be a strictly positive integer and consider the recursive sequence n N* an= a(n 1) 2 if a(n 1) is even 3a(n 1) 1 if a(n 1) is odd So there exist n N such that an 1. The proof Let (un)n>=0 be the subsequence of odd integers of the sequence (an)n>=0. So there exist a sequence of strictly positive integers (an)n>=1 such that n N* 3u(n 1) 1 2an un Since un is odd we have un 2vn 1 with vn N for each positive integer n. Let n N* bn an 2 bn is an integer with bn 1. Thus n N* 3(2v(n 1) 1) 1 2bn+2(2vn 1) So n N* 6v(n 1) 4 2bn+3 vn 2bn+2 So n N* 32 v(n 1) 1 2bn+1 vn 2bn So n N* 32 v(n 1) 1 2 x 2bn vn 2bn (1) Let N N* from equality (1) we deduce that 32 v0 1 2 x 2b1 v1 2b1 32 v1 1 2 x 2b2 v2 2b2 x 2b1 32 v2 1 2 x 2b3 v3 2b3 x 2b1 x 2b2 32 v(N 1) 1 2 x 2bN vN 2bN x 2b1 x 2b2 x x 2b(N 1) We multiply each equality by the number on the right we get

title: a Proof of the Collatz Conjecture
abstract: The Collatz Conjecture has attained notoriety in the Mathematics community for being a problem easy to state and understand, but is apparently difficult to solve. Arguably, the largest barrier to a solution was given by Conway's proof on certain generalizations of the collatz mapping being uncomputable. This paper will present a novel Collatz function and prove some properties of it, it is also written with the explicit purpose to make the proof short and easy to understand with basic knowledge.

title: Paths in primal spaces and the Collatz conjecture
abstract: The Collatz conjecture establishes that for every natural number n N, there exists an r N such that kr (n) 1, where k N N is the function defined as n/2 if n es even and as 3n 1 if n is odd. The map k induces a topology tk on N. We prove that the Collatz conjecture and connectedness of the space (N, tk) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (N, tk) is path connected.