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 ...

learn more… | top users | synonyms

76
votes
8answers
17k views

What is the importance of the Collatz conjecture?

I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem: Take any natural number $n$. If $n$ is ...
22
votes
4answers
770 views

Why is $3$ the coefficient of $n$ in the Collatz conjecture?

What's the importance of multiplying a 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 ...
13
votes
3answers
1k views

The $5n+1$ Problem

The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows: Define $f(n)$ to be as a function on the natural numbers by: $f(n) = n/2$ if $n$ ...
9
votes
2answers
540 views

Thoughts on the Collatz conjecture; integers added to powers of 2

I've had a thought about the Collatz conjecture (the 3n+1 problem). Suppose some number, C, diverges under the iteration. We first note that C must be odd because if C were even it would be halved ...
8
votes
4answers
1k 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 ...
8
votes
1answer
510 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 ...
6
votes
2answers
580 views

Longest known sequence of identical consecutive Collatz sequence lengths?

I've just written a simple java program to print out the length of a Collatz sequence, and found something I find remarkable: Consecutive sequences of identical Collatz sequence lengths. Here is some ...
6
votes
1answer
131 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
1answer
380 views

Required reading on the Collatz Conjecture

I am currently writing a paper on 3x+1 and realized that despite having enough knowledge to work on a singular facet of the problem I lack a more broad understanding of the problem. I have seen the ...
5
votes
2answers
115 views

How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
5
votes
2answers
348 views

What are some reasonable things to prove about the Collatz Conjecture?

I am writing an undergraduate paper on the $3n+1$ problem, and I am looking for some theorems related to the problem that would be reasonable for someone with my mathematical background to prove. I'm ...
5
votes
2answers
209 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
5
votes
2answers
354 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 ...
5
votes
3answers
75 views

How can I prove that an iterated transformation describes all odd integers?

This is a question where I want to find "a best" way (or even different ways) to prove my assumption - just to widen my understanding of similar problems and how to approach them. It's a question of ...
5
votes
1answer
105 views

What is the simplest collatz like problem that is undecidable?

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 ...
5
votes
0answers
200 views

Collatz-like : $(5 x \pm 1)/3$ [closed]

For a motivating start, recall the definition of the usual Collatz iteration on the integers (binding exactly one halving with each tripling), rewritten in a (slightly) unusual way. The Collatz ...
4
votes
1answer
233 views

Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.

For what values of $m,n$ natural, do $2^n - 1$ is divisible by $2^{m+n} - 3^m$? Thank you very much.
4
votes
3answers
60 views

Examples of “eventually reaches y under iteration” other than the Collatz problem

The Collatz conjecture states that iteratively applying the map $$f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ to any ...
4
votes
2answers
112 views

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$?

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ? (i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)? Let the number ...
3
votes
2answers
153 views

About the Collatz conjecture

I worked on the Collatz conjecture extensively for fun and practise about a year ago (I'm a CS student, not mathematician). Today, I was browsing the Project Euler webpage, which has a question ...
3
votes
2answers
298 views

Prerequisite reading before studying the Collatz $3x+1$ Problem

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 ...
2
votes
2answers
115 views

Where can I find out more on Collatz-conjecture like sequences?

I'm interested in Collatz-conjecture (the 3n+1 problem) like sequences. I'm interested in any literature that contains information about problems that are divided into similar cases. I'm ...
2
votes
3answers
288 views

Is this statement stronger than the Collatz conjecture?

$n$,$k$, $m$, $u$ $\in$ $\Bbb N$; Let's see the following sequence: $x_0=n$; $x_m=3x_{m-1}+1$. I am afraid I am a complete noob, but I cannot (dis)prove that the following implies the ...
2
votes
1answer
74 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 ...
2
votes
0answers
68 views

arithmetic sequence $8n+1$ and the collatz conjecture

Is it a known result that if for all $n$ the collatz sequence of $8n+1$ lead to $1$, all natural numbers will?
2
votes
2answers
52 views

My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
2
votes
1answer
64 views

Diophantine equation $3^{m}x+3^{m}-3^{m-i} 2^{i} +3^{m-i} 2^{i+s} -2^{m+s}=x 2^{m+s}$

As part of my research and my calculations I got the following diophantine equation. I do not have much experience with the diophantine equation. Is there any known method to solve it? Any help is ...
1
vote
4answers
579 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 ...
1
vote
2answers
228 views

$N \equiv 3 (\textrm{mod } 4)$ and Collatz conjecture

Can the Collatz conjecture also be interpreted as behaviour, transformation of number of form $N\equiv 3(\textrm{mod }4)$ to the form of $N\equiv 1(\textrm{mod }4)$ Because integers of the form ...
1
vote
3answers
561 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 ...
1
vote
2answers
466 views

About Collatz 3n+3?

While trying to prove the Collatz conjecture I came across the following Lemma : Lemma $1$ : All variables are positive integers. Define $Collatz(n)$ as the result of the (repeated) collatz ...
1
vote
3answers
454 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 ...
1
vote
1answer
396 views

What is known about Collatz like 3n + k?

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 $3n+3$ ...
1
vote
3answers
78 views

For how many consecutive numbers Collatz conjecture was checked?

I heard here that Collatz conjecture was checked at least for every first $5 \cdot 10^{18}$ natural numbers, but I cannot find any source or actual information about this. Can anyone help to find out ...
1
vote
1answer
110 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
3answers
94 views

Collatz/Ulam/3n+1 with 2D array approach [closed]

Have there been any attempts on the 3n+1 problem using the fairly obvious infinite 2D array ...
1
vote
1answer
28 views

Modified Collatz Problem

How can one prove for all $n \in \mathbb N$ that the following sequence always results in $1$: Choose $m$ $x_1 = m$ $$x_{n+1} = \begin{cases} x_n/2 & \text{if $x_n$ is even} \\ \\ x_n + 1 & ...
1
vote
1answer
65 views

Proof of Terras' method for finding T(k)(x) of Collatz sequence.

Terras (via Lagarias) has a method explained here (2.2) for finding the kth term of any Collatz sequence (that is) $$T(n) = \frac{3n+1}{2}$$ if n is odd and $$=\frac n2$$ if n is even. The lambda ...
1
vote
0answers
98 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 ...
1
vote
1answer
276 views

Converse of Collatz Conjecture

How to write a pseudocode program that halts only if the Collatz Conjecture is false. Thanks much in advance!!!
0
votes
3answers
186 views

I have a proof of the Collatz conjecture . Where should I submit my proof?

I have been working on the Collatz conjecture starting around 10 years ago. I would like to submit my proof for analysis. I would like advice on where to submit my proof.
0
votes
1answer
4k views

What does the right arrow mean in the example provided.

My second question here. Does it show that I know very little about mathematics? :) I'm doing Project Euler Question 14 and would like to know what the right arrow → means in: ...
0
votes
1answer
58 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, ...
0
votes
1answer
66 views

It's sufficient to prove collatz conjecture for $3+6k, k \geq 0$?

Thinking about this problem, I saw two interesting properties of Collatz graph. Firstly, if we consider that every even number $e$ can be represented (on a single way) as $e = o 2^n$, where $o$ is an ...
0
votes
1answer
112 views

collatz conjecture $\mod 2^n$ stopping distance

an interesting book Old and new unsolved problems in plane geometry and number theory by Klee/Wagon (1991) includes the Collatz conjecture. on p225 they consider iterates $\mod 2^n$ and state that ...
0
votes
1answer
28 views

Given a natural number $a$ find its index in a set of structural descriptions

Looking at orbits of the collatz-like $(5x+1)/2^X$ - map I come to a useful structural description for all odd integers $a$. If I write $$ {5a+1 \over 2^A} \to b \qquad \qquad \text{for odd positive } ...
0
votes
1answer
83 views

Are there any explanations for these patterns in the Collatz sequences?

I've been messing around with the Collatz sequences a bit, and have come across a few patterns - I was wondering if there are any known explanations for these patterns. The first is the plot of ...
0
votes
1answer
89 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
-1
votes
2answers
645 views

What to do with Collatz proof? [closed]

I have discovered a proof solving the Collatz problem, and I have no idea what to do with it. Given the nature of the topic, all the experts I've found that are capable of reviewing the paper ...
-4
votes
1answer
122 views

A contraction approach on Collatz conjecture [closed]

Talking about Collatz like iterative sequences, If we define Collatz function as $f(x)=k(2.5x+1)+x/2$ where $k=x$%2 then according to Banach fixed point theorem(from wiki), "Every contraction ...