Questions tagged [collatz-conjecture]

For questions about the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even, and its generalizations.

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1answer
986 views

When is a function a permutation of the integers?

When is a function a permutation of the integers? In his 2011 paper on the Collatz conjecture here Lagarias writes; ...
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1answer
117 views

Collatz conjecture

See this After seeing this question, I observed first 10 natural numbers , I saw this For every $n\in \mathbb N$ and $n\ne 2^k$ for some $k\in \mathbb N$ , after applying these two operations , ...
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1answer
127 views

Collatz Conjecture and the Set Theory [closed]

Let us define the two functions $F(n)$ and $G(n)$ as given: $$ F(n) = \begin{cases} 3n+1 & \text{ and then dividing through by any powers of $2$} \end{cases} \\ G(n) = \begin{cases} 3n-1 &...
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1answer
218 views

How well do prime-started Collatz sequences cover the integers?

Consider a liberal definition of the Collatz sequence starting from some prime number $p$: $$ C_n(p) = \left\{ \matrix{p & n=0\\C_{n-1}(p)/2 & C_{n-1}(p) \mbox{ even} \\ 3C_{n-1}(p)+1 & ...
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3answers
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Collatz conjecture pattern (3n + 1 problem).

I have a pattern I found in Collatz Conjecture I want to share. Afterwards, I would like to know, if I could try harder at this pattern (I am stuck), if it could lead to a proof. Or it would just be ...
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1answer
644 views

If the Collatz Conjecture is unsolvable is it true?

Recently Numberphile uploaded a video on Godel's Incompleteness Theorem, and the Professor in the video made the conclusion that if you can prove a statement cannot be proven true or false by the ...
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2answers
201 views

Why does the expression $3n+1$ appear in Collatz conjecture?

Why is it important the role of $3n+1$ in Collatz conjecture? I mean, if we replace $3n+1$ by $5n+1$ it seems (numerically) that the modified statement of Collatz conjecture does not hold in this case....
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2answers
350 views

Why is this wrong?

About the Collatz Conjecture: Every body looks at it from "leaves to root" - to use the tree analogy. I have another approach. My approach is to look at it from the root - the number 1 - and see if, ...
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1answer
95 views

Disprove this. Please.

About the Collatz conjecture: Let Steps be: Number of steps, S, that a "counting number", n, takes to reach 1 - sometimes referred as the stopping time (of n); Example: 5 take 5 steps to reach 1 (...
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1answer
285 views

In the Collatz conjecture, why are $\max(\textrm{collatz}(n))$ and $\textrm{var}(\textrm{collatz}(n))$ so closely related?

Like the question title reads, in the Collatz conjecture, why are $\max(\textrm{collatz}(n))$ and $\textrm{var}(\textrm{collatz}(n))$ so closely related? See the Figure below for a log-log plot. I ...
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1answer
668 views

How many “non-Collatz-numbers” do exist?

Everybody knows or at least heard about Collatz or $3x+1$ conjecture. Let us now define something like: Definition 1: Number $m \in \mathbb{N}$ is called $k-Collatz$ number if in its sequence ...
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1answer
587 views

What is the latest verified research on the 3x+1 Problem? [closed]

Wikipedia : Collatz Conjecture Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which ...
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1answer
161 views

Can there exist such a Collatz cycle that never divides by 2 more than one time between the increasing parts?

Sorry for the long title. I first of all want to say that I'm just a high school student who spent today looking into the Collatz conjecture. I, first of all, would like to know if it's known whether ...
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1answer
91 views

Approaches to solve a collatz-ish function

Let's say we've a function similar to the function in Collatz conjecture. $$ f(n)= \begin{cases} 1 \ \ \ \ \ \ \ \ \ \text{if $n=1$}\\ \tfrac12n \ \ \ \ \ \ \text{if $n \equiv 0 \ \ $ (mod 2)}\\ n-1 ...
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1answer
84 views

A very different property of numbers: Can be changed to $1$ by applying only these two operations.

While playing with numbers, I thought of type of numbers, and then the first thing came into mind was $\text{Odd}$ and $\text{Even}$. I observed a very interesting fact that any $x\in\Bbb{N}$ can be ...
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2answers
503 views

Is it possible to find general formulas that can express all collatz numbers?? [closed]

For collatz hypothesis, is it possible to find general formulas which will give 1 result ?? Is it possible that we can find all collatz numbers with these formulas? Have the generic formulas been used ...
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1answer
275 views

On a Collatz-like problem with two end cycles

In an alternative version of Collatz problem, one iterates the function $f:\mathbb{N}\to\mathbb{N}$ defined by $$f(n)=\begin{cases}n/3 &\mbox{if}\ n\equiv0\ (\mbox{mod}3)\\ 2n+1&\mbox{if}\ n\...
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0answers
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Is there a proof for this modified Collatz-like problem?

The Collatz Conjecture is a famously unproven problem in mathematics, but I was thinking of a slight modification, and whether or not a proof of this different form is trivial. Here is a statement of ...
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0answers
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Property unique to Collatz?

Just an amateur having fun with some sequences and would like to understand what I'm seeing. Playing with Collatz in python and found something interesting when I plotted the following: Where I plot ...
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2answers
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Smallest number of a hypothetical second Collatz Cycle

Most here are probably aware of the Collatz Conjecture. It is conjectured that every number eventually ends in a trivial cycle of 1 -> 4 -> 2 -> 1 if you follow these rules: Take any number: If it ...
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1answer
86 views

Find an infinite sequence of statements so that truthness of all sequence elements is unprovable

Let us consider a function $f: \mathbb N \to \left\{0, 1\right\}$ with finite description length (i.e. describable by a finite length program for a Turing machine) satisfying the following conditions: ...
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1answer
181 views

The connection between the Collatz Conjecture and the Optic Equation

The Collatz Conjecture is fun for those of us who have never taken mathematics in university in any meaningful way. It's also probably fun for all of you who have gone to university. So I want to ...
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2answers
97 views

Can all functions be theoretically iterated succesfully?

I'm trying to solve the Collatz conjecture, and am having some trouble designing a function that divides a number by two until it's odd. Here is what I've thought of. We know $a\mod b = \arctan(\tan(...
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4answers
191 views

Biggest factor of 2 of any number N

I'm trying to get an insight of the Collatz Conjecture (3n+1 conjecture), and have been researching about iterated functions and thought the conjecture could possibly be solved if I was able to ...
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1answer
339 views

Collatz divide by -2 instead

I've been toying around with the Collatz conjecture for a while, and in an effort to extend it to the negative integers I tried diving by $-2$ instead of by $2$. The new iteratively applied function ...
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1answer
145 views

Must any claimed proof that the nonexistence of non-trivial cycles in the Collatz Conjecture is unproveable, be false?

If $C$ is the nonexistence of non-trivial cycles in the Collaz conjecture, and $CC$ is the Collatz conjecture itself. If it were proven that $C$ were unproveable, would this mean that $\lnot C$ would ...
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0answers
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Alternative notation for the Range of these functions

I have n functions with n variables: $x_1, \dots , x_n \in \mathbb{N}/0$ that follow a nice pattern: $f_1(x_1) = \frac{2^{x_1}}{3} - \frac{1}{3}$ $f_2(x_1,x_2) = \frac{2^{x_1+x_2}}{9} - \frac{2^{...
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1answer
184 views

Collatz Number System Question

I was looking at the Collatz Conjecture and I thought of something: If we denote two operators $a_n = 2n$ and $b_n = \frac{n-1}{3}$, then every number that converges using the Collatz Conjecture can ...
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2answers
137 views

Does $\frac{(3x+1)}{2^n}$ generate a partial ordering of the odd, positive integers?

Taking $n,m\in\mathbb{N}$ and $n,m >0$ throughout. Does the graph of the function $f(x)=\frac{(3x+1)}{2^n}$ generate a partial ordering of the odd integers? Because if it did, it would seem to me ...
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1answer
471 views

Intuition behind lack of cycles in the Collatz Conjecture

The Collatz Conjecture concerns the function $f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$ . The conjecture says that if you start with ...
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1answer
111 views

The square of every prime number can be expressed in a linear form. [duplicate]

While working on the Collatz conjecture, I've found that the square of every prime number $p$ (except 2 and 3)can be written in the form of $12k+1$. $p^2=12k+1$.$(k\in\mathcal N)$ is this a new ...
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1answer
577 views

Is this a proof of the Collatz Conjecture?

I recently stumbled upon the following paper from April 2016: https://www.researchgate.net/publication/299749569_A_proof_of_the_Collatz_conjecture Its researchers, who are university professors, ...
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3answers
220 views

Regarding Collatz, ratio of odd steps to even steps?

If $b > 1,\ \ b \ \to_c \ 1$ in $m + n$ steps ($m$ through odd numbers, $n$ through even) then $2^A + b \ \to_c \ {3^m \over 2^n} 2^A + 1$. If we imagine that ${3^m \over 2^n} 2^A + 1 < 2^...
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1answer
3k views

Highest proven Collatz Conjecture stopping time [closed]

I'm working on a code challenge that requires finding the total stopping time of a number. I chose a recursive solution because it was the smallest, but Clojure can't guarantee tail-call optimization ...
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2answers
70 views

Patterns of factorization numbers of form $3/2 x +1/2$ for odd x?

I was doing some research into the Collatz conjecture and the main problem as I see it is this: What does +1 do to a factorization? Is there any patterns, theorems or recent research that tackles ...
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1answer
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Attempting Collatz conjecture using inverse function

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{...
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1answer
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Collatz Conjecture - Programmed in Python with Inf Result

Having written the following program in Python: ...
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1answer
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A bound regarding the Collatz conjecture from Wikipedia [closed]

Recently, I have been reading about the Collatz Conjecture and on the wikipedia page for it, came across the fact that: Rigorous bounds Although it is not known rigorously whether all positive ...
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5answers
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Collatz Conjecture (3n+1) variant

Let's consider the following variant of Collatz (3n+1) : if $n$ is odd then $n \to 3n+1$ if $n$ is even then you can choose : $n \to n/2$ or $n \to 3n+1$ With this definition, is it possible to ...
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1answer
62 views

(More efficiently) solving for residue classes $\pmod {2^A}$?

This is in context to a detail for the generalized Collatz-problem (generalized to various multipliers, like 3x+1, 5x+1, 7x+1, ... ,mx+1, ...) I am currently looking at the 11x+1 problem , and as a ...
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2answers
82 views

What does this strange limit indicate?

I have been evaluating limits of the Collatz and Waring sequences and have found one strange result (top line). For all of the others, $-\infty, +\infty$ produce equal results. $$ \lim_{n\to -\infty }(...
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1answer
362 views

On the Collatz Conjecture with odd multiplier of five and with the starting value, n_0 = 7. [duplicate]

In the Collatz Conjecture, if the odd multiplier is changed from three to five, what is the outcome for a starting value of seven? Does the Collatz sequence converge to one or diverge? (Reference ...
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3answers
2k views

The Collatz conjecture algorithm applied to negative integers

Does applying the Collatz conjecture algorithm to negative integers always result in sequences ending with $-1$ and then repeating themselves an infinite number of times? It is so for the first ...
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1answer
136 views

Is it known that there are an infinite number of odd solutions for the Collatz conjecture?

Take all odd numbers that are non-divisible by 3. These are in the form of 2(3n)+1 and 2(3n+2)+1. Simplify: 2(3n)+1 6n+1 Find 3n+1: 3(6n+1)+1 9n+2 Simplify: 2(3n+2)+1 6n+5 Find 3n+1: 3(6n+...
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2answers
59 views

Knight moves on a Triangular Arrangement of the First Iteration of the Collatz Function

This is related to the Collatz function which can be written $$T(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ All I did was ...
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1answer
395 views

Is there a number that using the rules of Collatz conjecture's variation $3n-1$ doesn't get to $1, 7$ or $17$?

The rules are simple: Take any number $n$. If $n$ is even divide it by two, if $n$ is odd triple it and subtract one. Repeat indefinitely. (Note that this is a variation, in the original Collatz ...
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2answers
543 views

Simplified variant of Collatz conjecture.

I came across the Collatz conjecture. So apparently the idea is to see if all prime factors of a number can be 'annihilated' by successive steps of either removing a factor of two, if n is even or in ...
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1answer
201 views

Collatz Patterns

I have seen documentation on the $4K+1$ pattern, but as of yet I have seen nothing on the $64K+35$ pattern or the $262144K+184471$ pattern. Is there anywhere I can read up on these? I created the ...
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2answers
165 views

What facts are known about the hypothetical smallest divergent integer in the collatz conjecture?

If there is a divergent integer in the collatz conjecture then there must be a smallest divergent number by the WOP. We can observe some properties of this number such as it must be odd because if it ...
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1answer
409 views

What level of mathematics do I need to study the Collatz Conjecture?

I recently came across the Collatz Conjecture and I'm really intrigued by its tautological simplicity and complexity. I'm under no illusions that I can make any progress with a proof for it but I ...

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