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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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
399 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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567 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
202 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
167 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
420 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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251 views

Will the Collatz conjecture work for $m \cdot (n)+1$ for an odd number, where m is any odd number?

The Collatz conjecture asks you to: When '$n$' is the given number, 1) Divide $n$ by $2$ if the number is even. 2) Do $3n+1$ when the number is odd, and you will reach the series $4->2->1$. ...
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How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
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1answer
730 views

On solving the Collatz conjecture

This method may be kinda inefficient as solving each step may require $O(n!)$ computational time, but for $n$ Collatz operations isn't it possible to disprove the existence of a cycle of $n$ ...
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419 views

Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes ...
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Collatz Conjecture proof that no cycle can exist other than the 1,4,2 cycle. Can someone verify it?

This is not a proof of the Collatz Conjecture, but I somehow managed to show that there is no cycle that can exist other that the 1, 4, 2 cycle: $n$ is a positive integer $3n+1$, $n$ is odd $\frac ...
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201 views

Has it been proven that there is no closed form for the hailstone numbers?

I know none has been found, and there probably isn't one considering the effort people have put into it, but has it been proven? (for some reasonable definition of "closed form"). I'm mostly ...
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4answers
391 views

List of properties of the Syracuse sequence

Where can I find an as exhaustive as possible list of all the properties (empirical or proven) related to the Collatz conjecture ? For example I noticed that starting from $2^{n}-1$ the sequence ...
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1answer
87 views

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Let the $3x+1$ map $$ f(n) = \begin{cases} 3n+1 & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} .$$ Now we read the Wikipedia's page for the Collatz ...
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Behavior of a Collatz-like mod-4 sequence: Do some numbers increase without limit?

Define \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \end{eqnarray} and let $f^k(n) = f(f( \cdots (n) \cdots )...
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Collatz Conjecture, sufficient to show odd numbers reach $1$?

The famous conjecture: Let $$ f(n) = \begin{cases} n/2 & \quad \text{if } n \text{ is even}\\ 3n+1 & \quad \text{if } n \text{ is odd}\\ \end{cases} $$ The Collatz Conjecture ...
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1answer
383 views

Do all primes occur in some sequence associated with the Collatz conjecture?

Let $f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$ For an arbitrary prime $p$ are there some start value $x_0$ such that $p = x_k$ for some ...
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1answer
789 views

Can it be proven that Collatz numbers cannot repeat?

One potential counterexample of the Collatz conjecture would be if there was a number that looped back to itself. Of course, this would still not prove the conjecture because some sequences could ...
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1answer
131 views

How to prove equally likely steps of the Hailstone sequence (collatz sequence)

Consider the condensed collatz conjecture if $x$ odd then $f(x)=(3x+1)/2$: if $x$ even $f(x) = x/2$: Continue until $x = 1$ or find an $x$ in the natural numbers that will not hit $1$. The equation ...
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1answer
97 views

Closed form of iterated function

It can be easily calculated that performing the operation $(3x+1)$ on a number m, k times yields the result $(3^k)(m)+(3^k -1)/2$ I want to calculate a closed formula for performing the related ...
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661 views

hailstone sequence of perfect squares (collatz conjecture)

The Collatz conjecture states: 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 ...
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1answer
36 views

Don't know what this expanding periodic-ish function is

I plotted a function $c(x)$, which returns $3x + 1$ if $x$ is odd, and $x/2$ if $x$ is even. It's the Collatz conjecture. I get this interesting function. I don't know what it's called, so I can't ...
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1answer
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How many steps to reach 1? (Collatz Conjecture) [closed]

Is there some sort of algorithmic process or equation to determine the number of steps required for any given integer n to reach 1 in the Collatz Conjecture without having to actually perform a ...
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5answers
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Is this a proof for the Collatz conjecture

For this problem, which I believe is still unsolved, I was wondering what is wrong with this proof I thought of (probably is wrong somehow) https://en.wikipedia.org/wiki/Collatz_conjecture So my ...
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1answer
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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) = \...
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How is this fractal produced?

It is stated here: Iterating the above optimized map $$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$$in the complex plane produces the Collatz fractal. The point of view of iteration on ...
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Extension of the collatz function to $\mathbb{C}$

The 3x+1 map is give as $$f(x) = \begin{cases} \frac{3x+1}{2} & \text{ if x odd} \\ \frac{x}{2} & \text{ else} \end{cases}$$ with domain $\mathbb{N}.$ On this wikipedia article, I found ...
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What would be the implications of (dis)proving the Collatz conjecture? [closed]

The Collatz conjecture is one of the most famous unsolved problems in mathematics. It essentially states that, for any positive integer, if you repeatedly apply the function ...
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1answer
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How can I generalize this?

I'm not sure what to tag this question as or whether its a bit nonsensical, but I'm a bit curious. I asked a question on a pretty (turned out to be) easy question about the Collatz Sequence here: ...
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260 views

Applying Collatz function iterations to large integers

Given the Collatz function and its iterates: $$ T(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ (3n+1)/2, & \text{if $n$ is odd} \end{cases} $$ and $$ T^{k}(n) = T(T^{k-1}(n)).$$ ...
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The longest known cycle length of generalized collatz $5x+1$ trajectory

The generalized collatz $5x+1$ trajectory, if $n$ is even then divide $n$ by $2$, and if $n$ odd then multiply $n$ by $5$ and then add $1$. For example if $n=3$, we have $3=>16=>8=>4=>2=&...
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Prime Factorization patterns of $\sum_{i=0}^k{4^i}$

$$ \text { Given the function: }f:\mathcal{N^+} \to \mathcal{N^+} where f \left(k\right) = \sum_{i=0}^k \,4^i. $$ Examining the prime factorizations of f(k) for k= 1...48, many factors appear in a ...
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Can Collatz's problem be used as a pseudo random prime sieve?

If you take the concept of $3x+1$, $\dfrac{x}{2}$ and starting at 2, create a tree. On the left nodes you apply the $3x+1$. On the right nodes, if the parent node is even apply the $\dfrac{x}{2}$. ...
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Partitions of the odd integers

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 ...
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548 views

(Collatz) Modulo 18 Partitions of Collatz 3n+1 Trajectories

I have examined partial Collatz 3n+1 trajectories going from one odd integer to the next. These lead to an infinite number of repeated patterns where the "next" odd integer is congruent to one of ...
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Collatz $4n+1$ rule?

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 ...
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Collatz Conjecture Algorithm

Related: A general question about the Collatz Conjecture and finding that integer that doesn't work I was coding a solution to the $3k+1$ problem and was looking at ways to speed up the computation. ...
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Generalizations of the Collatz to $(mx \pm 1)/2$ for $m=181$ gives two nontrivial cycles; are more examples $m$ known?

Generalizing the Collatz $T_{3,+}(n) = \left\{ \begin{array} {cl} {3n+1 \over 2} & \text{ when } n=2k+1 \\ \frac n{2^B} & \text{with maximal } B \gt 0 \text{ where } 2^B | n \end{array} \...
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Can any functions be expressed in one formula? (without conditions)

This is the extension of my previous inquiry: Is it possible to describe the Collatz function in one formula? Can each of all functions be expressed in one formula? That is, can any function ...
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Is the 3x+1 problem solved? [closed]

I found an article by Peter Schorer from June 29,2015 which is claming to give a solution of the 3x+1 problem. Are there remarks from any mathematicians if this is correct or not?
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Collatz-type problems with known divergent trajectories

It is a well-known problem, known as Collatz problem, to determine whether iteratively applying the map $f(x)=\frac{x}{2}\text{ if $x$ is even and }f(x)=3x+1\text{ otherwise}$ on a positive integer $n$...
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Does this extension of the Collatz sequence converge for n=550?

The Collatz function, or $3n + 1$ function is well known. A heuristic argument that most inputs should converge with repeated application of the function is as follows. With probability 1/2 an input $...
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Is it possible to describe the Collatz function in one formula?

This is related to Collatz sequence, which is that $$C(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ Is it possible to describe ...
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Could this odd insight help explain part of the difficulty in proving the Collatz Conjecture?

Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You ...
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In the $x + 1$ problem, does every positive integer $x$ eventually reach $1$?

I know that the more famous $3x + 1$ problem is still unresolved. But it seems to me like the similar $x + 1$ problem, with the function $$f(x) = \begin{cases} x/2 & \text{if } x \equiv 0 \pmod{2}...
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239 views

Is the Collatz function piecewise linear?

I read somewhere that the Collatz function $\mathbb Z \rightarrow \mathbb Z$: $$\text{Collatz}(x) = \begin{cases} x/2 &&x \; \mathrm{even} \\ 3x+1 &&x \; \mathrm{odd}\end{cases}$$ is ...
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Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
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Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map $m\...
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726 views

Uses of “Collatz induction”?

The Collatz conjecture is equivalent to the following "induction principle": If $P(0) \land P(1) \land (\forall{x} P(3 \cdot x + 2) \implies P(2 \cdot x + 1)) \land (\forall x P(x) \implies P(2 \cdot ...
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Looking for a verification or refutation my attempted proof of why the Collatz conjecture is probably false. [closed]

Most people think that the Collatz conjecture is true, but I think that I can prove the opposite. Let's make two functions, $f(x)$ and $g(x)$. $f(x) = $ The amount of numbers that can be solved in x ...

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