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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Outline approach to Collatz 3n+1 conjecture / Criticism needed

This is a sketch of an approach to proving the Collatz 3n+1 conjecture true along the following lines. Instead of trying to show there are no loops and no sequences that increase without bounds, ...
3
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0answers
57 views

Is this a valid statement that would imply the Collatz Conjecture?

Let $f$ denote the Collatz transformation: $f(x) = \left\{ \begin{array}{ll} {x\over 2} & \quad x\equiv 0 \mod 2 \\ 3x+1 & \quad x \equiv 1\mod 2 ...
1
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1answer
44 views

Simon Letherman, Dierk Schleicher, and Reg Wood 1999 paper on the 3n+1 problem, further results

I recently read with great interest the 1999 paper : "The 3n+1-Problem and Holomorphic Dynamics Simon Letherman, Dierk Schleicher, and Reg Wood" Experimental Mathematics 8: 3 (1999), 241–252. I ...
5
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2answers
133 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 ...
3
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2answers
98 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?
0
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1answer
79 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 ...
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3answers
88 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 ...
0
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3answers
288 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.
4
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3answers
64 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 ...
0
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1answer
61 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, ...
2
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1answer
83 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 ...
0
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1answer
30 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 } ...
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1answer
31 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 & ...
0
votes
2answers
106 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 ...
2
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2answers
55 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, ...
1
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1answer
75 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 ...
4
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1answer
238 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.
2
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1answer
73 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 ...
0
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1answer
90 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 ...
2
votes
2answers
117 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 ...
5
votes
1answer
110 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 ...
1
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1answer
125 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 ...
0
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1answer
123 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 ...
3
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2answers
172 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 ...
6
votes
2answers
664 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 ...
22
votes
4answers
825 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 ...
5
votes
2answers
218 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 ...
6
votes
1answer
141 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 ...
5
votes
3answers
79 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 ...
1
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0answers
100 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 ...
5
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0answers
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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 ...
5
votes
2answers
370 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 ...
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1answer
495 views

Have 3n + 1 Problem Proof --But can't do the Mathspeak [closed]

I've come here as a last resort. When I first saw the Collatz conjecture I worked out pretty quickly what the dynamic was that drove the Collatz sequences. That was in 2009. Since then I've been ...
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1answer
280 views

Converse of Collatz Conjecture

How to write a pseudocode program that halts only if the Collatz Conjecture is false. Thanks much in advance!!!
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3answers
598 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
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3answers
492 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 ...
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2answers
238 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 ...
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 ...
1
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1answer
405 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 $3n+3$ ...
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2answers
478 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 ...
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2answers
673 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 ...
3
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2answers
305 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
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3answers
292 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 ...
8
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1answer
529 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
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1answer
388 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 ...
0
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1answer
5k 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: ...
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2answers
573 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 ...
10
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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 ...
5
votes
2answers
365 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 ...
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4answers
596 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 ...