Related papers: Using binary string to prove the Collatz conjectur…
We intend to contribute to the Collatz dynamics problem by seeking to analyze the Collatz conjecture from the tree of numbers sequences. First, we show numerically that the distribution of odd numbers has an initial transient, and proceeds…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
We developed an algorithm that easily goes from one odd number to the next odd number in binary representation for the reduced forward Collatz map (Syracuse function). The algorithm indicates when an odd number can grow or shrink to the…
The Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an…
The Collatz process is defined on natural numbers by iterating the map $T(x) = T_0(x) = x/2$ when $x\in\mathbb{N}$ is even and $T(x)=T_1(x) =(3x+1)/2$ when $x$ is odd. In an effort to understand its dynamics, and since Generalised Collatz…
The 3n+1, or Collatz problem, is one of the hardest math problems, yet still unsolved. The Collatz conjecture is to prove or disprove that the Collatz sequences COL(n) always eventually reach the number of 1, for all n belongs to N+ (all…
For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd,…
I want to show one possibility to proof the Collatz conjecture, also called 3n+1 conjecture, for any natural number N. For this, I limit my analysis on the direct odd follower of every natural odd number and show the connections between the…
In a previous article, we reduced the unsolved problem of the convergence of Collatz sequences, to convergence of Collatz sequences of odd numbers, that are divisible by 3. In this article, we further reduce this set to odd numbers that are…
The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer $ n $. If $ n $ is even then divide it…
The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial'…
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 Conjecture sequences increase and decrease in seemingly random fashion. By identifying and analyzing the forms of numbers, we discover that Collatz sequences are governed by very specific, well-defined rules, which we call cascades.
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…
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…
This work represents an in-depth study of the structural behavior of the Collatz sequences. We consider a finite arithmetic progression with a common difference is 2 and the number of terms in the sequence is equal to 2^n . After, we…
Some simple facts are proved ruling the Collatz tree and the chains of vertices appearing in it, leading to the reduction of the number of significant elements appearing in the tree. Although the Collatz conjecture remains open, these fact…