Related papers: Integer patterns in Collatz sequences
The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate…
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…
Considering all possible paths that a natural number can take following the rules of the algorithm proposed in the Collatz conjecture we construct a graph that can be interpreted as an infinite network that contemplates all possible paths…
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…
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…
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
Motivated by a recent work of Tr\"umper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application…
I show here that there are three different kinds of iterations for the reduced Collatz algorithm; depending on whether the root of the number is odd or even. There is only one kind of iteration if the root is odd and two kinds if the root…
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…
Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs. Our main result is an infinite family of counterexamples to…
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…
Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study.…
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…
This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences…
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…
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…
In this paper, we show that if the numbers in the range $[1,2^n]$ satisfy Collatz conjecture, then almost all integers in the range $[2^n+1,2^{n+1}]$ will satisfy the conjecture as $n \to \infty$. The previous statement is equivalent to…
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…
We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. We conjecture that the orbit of arbitrary positive integer always returns to 1, as in the case of Collatz function. The conjecture is supported by…
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…