相关论文: The Collatz function as an automorphic Cayley colo…
This paper proposes a formula expression for the well-known Collatz conjecture (or 3x+1 problem), which can pinpoint all the growth points in the orbits of the Collatz map for any natural numbers. The Collatz map $Col: \mathcal{N}+1…
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…
Exploring the Collatz Conjecture and changing the expression from 3n + 1 to 5n + 1, we found patterns in different sets of numbers. Some numbers reduce to one (as stated in the Collatz Conjecture), some might escape to infinity, and some…
The famous (3n + 1) or Collatz conjecture has admitted some progress over the last several decades towards the conclusion that the conjecture is true (i.e. that all Collatz sequences will eventually reach a value of one), but has stubbornly…
Andrei et al. have shown in 2000 that the graph $\boldsymbol{\mathrm{C}}$ of the Collatz function starting with root $8$ after the initial loop is an infinite binary tree $\boldsymbol{A}(8)$. According to their result they gave a…
By using properties of Markov homogeneous chains and Banach measure in $\mathrm{N}$, it is proved that a relative frequency of even numbers in the sequence of $n$-th coordinates of all Collatz sequences is equal to the number…
The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x +1$ problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of…
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)
The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…
The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually…
We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the…
An attempt to come closer to a resolution of the Collatz conjecture is presented. The central idea is the formation of a tree consisting of positive odd numbers with number 1 as root. Functions for generating the tree from the root are…
On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial…
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…
An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.
The Collatz graph is a directed graph with natural number nodes and where there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd. Studying the Collatz graph in binary…
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…
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…
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…
Consider a finite positive integer. If it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. This will give you a new integer. Following the procedure for the new integer, you will receive another integer. Repeat the…