Articles of collatz

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any positive integer n, if n is even, divide it by 2; otherwise multiply it by 3 and add 1. If […]

How could Collatz conjecture possibly be undecidable?

I wonder how the collatz conjecture could possibly be undecidable? Since let’s say it’s undecidable, then that means that no counter example can ever be found, and that to me seems to imply that non exist, and thus that it’s true?

Give me such a $“3n±Q”$ problem that we do not know a counter-example

Maybe, This question is stupid.But, I want to ask.Because, really I dont Know answer.This problem may be similar to others. My Question is: $$ f(n) = \begin{cases} Pn±Q & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} ,$$ and we can find such $k$ $$f^{k}(n)=1$$ Here $P,Q\in \mathbb{N}$ For […]

What is known about Collatz like 3n + k?

I wonder what is known about variations of Collatz where $3n+1$ is replaced by $3n + 2k + 1$ where k is a fixed positive integer. In the OP ‘ about Collatz $3n+3$ ‘ it was confirmed that $3n+3$ behaves like Collatz itself. About Collatz 3n+3? I wonder about other values of $k$.

Determining the Collatz Series as a Tree of $\forall\mathbb{N}$

I’m proposing a proof for the Collatz Conjecture; and should like to take answers in terms of validation or contradiction to the arguments proposed. The conjecture states, where; $$ T(n) = \begin{cases} 3n+1 & n\in\mathbb{N}_{odd} \\ n/2 & n\in\mathbb{N}_{even} \end{cases} $$ $T(n)$ will converge to $1$ for $\forall\mathbb{N}^+$ Attempting the proof as follows We base […]

Prerequisite reading before studying the Collatz $3x+1$ Problem

Let’s assume I am starting college and have just finished calculus. I’ve been reading a bit online about the Collatz $3x+1$ Problem and find it to be very intriguing. However, a lot of what I’m reading uses terms and techniques that I have not seen before. I’m wondering what prerequisite (text book) reading is required […]

What is the simplest collatz like problem that is undecidable?

I have read that problems resemblings collatz have been shown to be undecidable. Conway proved that apparantly but Im not sure if the proof was constructive. So I wonder : What is the simplest collatz like problem that is undecidable ?

Continuous Collatz Conjecture

Has anyone studied the real function $$ f(x) = \frac{ 2 + 7x – ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so on) with respect to the Collatz conjecture? It does what Collatz does on integers, and is defined smoothly on all the reals. I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text{$n$ […]

About Collatz 3n+3?

While trying to prove the Collatz conjecture I came across the following Lemma : Lemma $1$ : All variables are positive integers. Define $Collatz(n)$ as the result of the (repeated) collatz iteration $x/2$ for even , $3x+1$ for uneven halt at $1$. Define $Collatz2(n)$ as the result of the (repeated) iteration $x/2$ for even , […]

Required reading on the Collatz Conjecture

I am currently writing a paper on 3x+1 and realized that despite having enough knowledge to work on a singular facet of the problem I lack a more broad understanding of the problem. I have seen the thorough annotated bibliographies by Jeffrey C. Lagarias but I do not have the time to read most of […]