Related papers: Some contributions to Collatz conjecture

Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics…

Let $\sigma_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{\sigma_n})$ of starting numbers $s$ with finite stopping time…

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)

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 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…

This paper introduces prime holdout problems, a problem class related to the Collatz conjecture. After applying a linear function, instead of removing a finite set of prime factors, a holdout problem specifies a set of primes to be…

In the present paper, we are interested in classifying of Collatz sequences on based to the different behavior of these sequences when their lengths tend to infinity. A Collatz infinite sequence can be defined as an infinite ordered set of…

Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd,…

Let an odd integer \(\mathcal{X}\) be expressed as $\left\{\sum\limits_{M > m}b_M2^M\right\}+2^m-1,$ where $b_M\in\{0,1\}$ and $2^m-1$ is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to…

The purpose of this paper is to show three general formulas of three global characteristic coefficients of Collatz function. The Collatz function is defined by the following operation on an arbitrary positive integer if N is odd multiply it…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

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.

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 $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that…

The 3x+1 Conjecture asserts that the T-orbit of every positive integer contains 1, where T maps x\mapsto x/2 for x even and x\mapsto (3x+1)/2 for x odd. A set S of positive integers is sufficient if the orbit of each positive integer…

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the…

The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by 2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there…

The celebrated $3x+1$ problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula $f(x)+1$ with a unique fixed point. The resultant formula $f(x)$ is also found to…

In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…

On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $ be the total iteration steps, the odd iteration steps and the even iteration steps when $N$ iterates to 1(except 1)…