Dynamics of the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even. Generalizations to $n \to 3n-1 $ or $ n \to 5n+1$ or even to $n \to pn+q$ . Other names are "$3x+1$-problem","syracuse problem". If you have a question, please be specific to your detail. MSE is not ...

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107 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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+50

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 ...
2
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3answers
159 views

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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0answers
43 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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1answer
80 views

Collatz Conjecture? [closed]

I am a lover of Math, all kinds really it is a bit of a puzzle to me I'm always trying to learn something new or a new "puzzle" to try an solve for myself even though many times i just reach the same ...
2
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3answers
69 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 ...
1
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1answer
49 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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12 views

Tag systems to cyclic tag systems and turing completeness

Consider the 2-tag system Alphabet: {a,b,c} Production rules: a --> bc b --> a c --> aaa and stating words aaa...a halts. on ...
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2answers
158 views

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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2answers
73 views

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
74 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
49 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
55 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
31 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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1answer
58 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
28 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
107 views

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 ...
1
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5answers
128 views

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
358 views

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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1answer
86 views

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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0answers
32 views

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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1answer
87 views

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
62 views

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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3answers
96 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 $$ T^{k}(n) = T(T^{k-1}(n)).$$ How ...
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0answers
82 views

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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3answers
93 views

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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0answers
57 views

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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3answers
290 views

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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1answer
206 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 ...
7
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1answer
586 views

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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0answers
60 views

Collatz Conjecture: Literature on Convergence

Does anyone know of a paper showing that if all n converge, they must converge to unity for n>0. Else, any literature related to convergence properties would be appreciated. Thanks, Jordan
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3answers
129 views

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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2answers
126 views

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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2answers
50 views

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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4answers
3k views

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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62 views

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 $n$...
6
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1answer
85 views

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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6answers
2k views

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 ...
7
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2answers
337 views

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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1answer
89 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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117 views

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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1answer
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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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2answers
154 views

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 ...
6
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2answers
96 views

Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period?

This is much like the $3x + 1$ iteration, except that if $x$ is odd, you do $5x + 1$ [and $\frac{x}{2}$ if $x$ is even]. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, ...
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Has the Collatz Conjecture been proven to be unprovable? [closed]

This paper, from a peer-reviewed journal, purports to prove that the Collatz Conjecture is unprovable. If it’s valid, why has it not received more attention? If it’s invalid, what is the flaw, and ...
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1answer
111 views

What causes long sequences of consecutive 'collatz' paths to share the same length?

I asked Longest known sequence of identical consecutive Collatz sequence lengths? some time ago, but I don't feel like it really got to the bottom of things. See, in the answers lopsy find a sequence ...
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Collatz algorithm generalization try-out (Collatz k-algorithm)

(Text Updated 2015/09/16, please see edit comments for changes) Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of ...
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457 views

Trying mathematical induction with $3n+1$ conjecture

Collatz's Conjecture is also known as the $3n+1$ conjecture. Well I thought since the conjecture is dealing with natural numbers so we might as well try mathematical induction and see why it doesn't ...