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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6
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2answers
88 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, ...
1
vote
5answers
119 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 ...
20
votes
6answers
2k views

What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz ...
1
vote
1answer
51 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 ...
1
vote
1answer
70 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 ...
0
votes
1answer
44 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 ...
1
vote
1answer
30 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 ...
1
vote
1answer
56 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 ...
-2
votes
1answer
95 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
vote
1answer
27 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 ...
3
votes
1answer
350 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) = ...
5
votes
1answer
79 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 ...
1
vote
0answers
31 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 ...
0
votes
1answer
96 views

Repeating cycles in the $3n-1$ problem

While tracking sequences beginning with 1-to-3 digit integers, I have found 3 different repeating cycles in the $3n-1$ problem (similar to the Collatz Conjecture). They are 1, 2, 1..., 5, 14, 7, 20, ...
3
votes
1answer
70 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 ...
1
vote
1answer
57 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: ...
1
vote
3answers
92 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 ...
0
votes
3answers
124 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. ...
2
votes
2answers
121 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} ...
7
votes
2answers
392 views

Prime numbers in Collatz sequences

This question/request is twofold. First, if this is a stupid question or if it has been addressed before, please say so (bluntness is optional), and I will crawl back into my cave... My question: is ...
6
votes
3answers
553 views

Probability and the Collatz Problem

A long time ago I was messing around with the Collatz problem and I stumbled across a "proof" that the iterations will converge. I was too embarrassed to show anyone, and I recently just remembered ...
7
votes
1answer
581 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 ...
1
vote
1answer
200 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 ...
1
vote
1answer
233 views

Collatz conjecture and related problems - mathematical machinery

Collatz conjecture stands as an open problem. That leads me to believe that the conjecture cannot be resolved by elementary means. Which brings me to my question: What techniques/machinery from ...
1
vote
5answers
685 views

Collatz Conjecture exclusivity

I have been wondering if there are any numbers that exist only in their own string of the 3n+1 problem. I need to explain that better. Basically, when you follow the rules of the conjecture, you end ...
2
votes
0answers
78 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 ...
1
vote
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 ...
5
votes
3answers
289 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 ...
0
votes
0answers
54 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 $\dfrac{x}{2}$. ...
29
votes
4answers
1k views

Why is $3$ the multiplicative coefficient in the Collatz conjecture?

What's the importance of multiplying an odd number by $3$ and adding $1$, instead of just adding $1$? After all, if you add $1$ to an odd number then it turns into an even number. Here is a example ...
1
vote
0answers
55 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
25
votes
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 ...
1
vote
2answers
49 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 ...
0
votes
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 ...
7
votes
4answers
2k 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?
1
vote
0answers
60 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 ...
6
votes
1answer
81 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 ...
8
votes
1answer
756 views

What are possibilities to disprove the Collatz Conjecture?

I was thinking about the Collatz Conjecture yesterday, and as opposed to trying to prove it, I was considering what would make the conjecture false. There were only two cases I could think of: We ...
0
votes
4answers
860 views

Ways of disproving proofs of the Collatz Conjecture?

I jokingly suggested for someone to prove the Collatz Conjecture, and they came up with their own proof. I have no idea how to disprove proofs, so can anyone tell me either what is wrong with this ...
2
votes
1answer
143 views

Is this first order version of the Collatz conjecture decidable in peano arithmetic?

Let $\phi(x)$ be a first order formula in the language of arithmetic with one free variable $x$. Consider the sentence $\psi_\phi$, defined as: $$\phi(0)\wedge \phi(1) \wedge (\forall x \phi(x) \to ...
7
votes
2answers
328 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 ...
1
vote
0answers
92 views

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 ...
4
votes
1answer
236 views

Collatz cycle necessary condition.

Has it been established that a nontrivial m-cycle of the Collatz conjecture on the positive integers would require two consecutive raises (i.e., if $\{x_1, x_2, \ldots x_n\}$ is the odd positive ...
1
vote
0answers
38 views

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, ...
1
vote
3answers
776 views

$5n+1$, $3n-1$ problem, smallest repeating cycle and Collatz conjecture

Among the Collatz conjecture we have other "similar" problems that are solved and have repeating cycles. $5n+1$ has the repeating cycle $13, 66, 33, 166, 83, 416, 208, 104, 52, 26$, with a length of ...
28
votes
1answer
393 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 ...
1
vote
0answers
116 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 ...
-4
votes
2answers
152 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 ...
4
votes
2answers
1k views

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 ...
1
vote
1answer
109 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 ...