Collatz conjecture If the function $f(n)$ is applied recursively enough number of times on any positive integer $n$, then unity will always be reached. \begin{align*} f(n) &= \left\{ \begin{array}{ll} n/2 &\text{if }n \bmod2=0 \\ 3n+1 &\text{if }n \bmod2=1 \end{array} \right.\\ \strut\\ \end{align*} It is an unproven problem. This question explores various approaches to attack the […]

Understanding the nature of the odd integers is a necessity to prepare oneself to work on the unsolved problems in number theory, such as the Collatz $3n+1$ problem. I hope to demonstrate how the odd integers can be represented as a sequence of sets which fit together like a glove with infinite fingers. First, some […]

I noticed something about the Collatz Conjecture, (I was literally obsessed with trying to prove it). I of course have NO intention of trying to prove it, clearly it is beyond my reach and I hope not to offend anyone by what may be a nonsensical observation, but I was a bit curious. This is […]

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 […]

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?

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 […]

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 $k$.

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 […]

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 […]

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 ?

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