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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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 ...
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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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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 ...
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A general question about the Collatz Conjecture and finding that integer that doesn't work

I apologize if this question gets down-voted ahead of time. I've been working on the Collatz Conjecture all day with Python, because that is the language I'm most familiar with (I'm not a CS student, ...
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Collatz algorithm generalization try-out (Collatz k-algorithm)

Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of professor Lagarias about it. Everything was so interesting (and I ...
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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 ...
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A question about the $3n+1$ conjecture

So I know that if you take any even number $n$ that is a power of $2$ like $32 = 2^5,16=2^4$ or $64=2^6$ we will keep dividing by 2 until we reach 1. and so all the steps will be $\frac{n}{2}$ and we ...
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Consequences of Collatz Conjecture being true

Collatz conjecture has been conjectured for a long time and I think there are some evidence showing that it should be true. Similar to $P \neq NP$ conjecture, is there some interesting ...
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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 ...
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225 views

Is anything known about the size of the smallest number with stopping time $n$

Last couple of days I've been thinking about the Collatz conjecture, and now I wonder if any relation is known between $n$ and the smallest number with stopping time $n$. So for example, let's say ...
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In the Collatz function, why does $2^k-1$ reach $3^k-1$ after $2k$ steps, and could it be used to find divergent trajectories?

If you start calculating the Collatz function from an integer of the form $2^k-1$, you will reach $3^k-1$ after $2k$ steps. So, it is possible to pick a starting value that continuously zig-zags ...
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In the Collatz function, why $3^{2k}-1$ and $3^{2k-1}-1$ always share the same trailing trajectory?

Why are the trajectories always the same for numbers of the form $3^{2k}-1$ and $3^{2k-1}-1$ for the Collatz function? For example, let $k = 3$. So, $3^6-1 = 728$ and $3^5-1 = 242$. The trajectories ...
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1answer
98 views

How could Collatz conjecture possibly be undecidable?

I wonder how the collatz conjecture could possibly be undecidable? Since let's say it's undecidable, then that means that no counter example can ever be found, and that to me seems to imply that non ...
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431 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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200 views

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any ...
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202 views

Is this property of the Collatz sequence interesting?

As an amateur playing around with the Collatz conjecture, I've stumbled on something I haven't seen mentioned before, and that may or may not be noteworthy. Suggested by Gottfried Helms, here's a ...
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On a proof that “there are at least $F_n$ Collatz permutations of length $n$”.

Let $n, k \in \Bbb{N}$ and $F_n$ be the $n$th term of the Fibonacci sequence. Let $u$ be the map $x \to 3x+1$ and $d$ be the map $x \to \frac{x}{2}$. Let a type be a sequence of $u$'s and $d$'s. ...
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Alternate proofs for Collatz 1-Cycles

To follow up on my comment here, I present my proofs of the Collatz 1-Cycles. I have asked to make this cw so others can edit as they wish. Here we show that the differences are equal to $t-1:$ ...
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The nonexistence of the Collatz-“1-cycle” by an elementary proof - am I missing something?

The so-called "1-cycle" in the Collatz-problem was already disproved by Ray Steiner 1977. However, he used transcendental number theory to achieve that, and Lagarias commented, it is surprising that ...
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What do these contour maps tell me about my Collatz expression?

I tested this limit on WolframAlpha, $$ \lim_{t\to\infty}\frac {2\ 3^r (2 t - 1) - 6} {3\ 2^r (2 t - 1) - 6}=\left(\frac{3}{2}\right)^{r-1},$$ which displayed two contour maps: . Can ...
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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 ...
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How to find the integer solutions of $\frac{2^m-1}{2^{m+x}-3^x}=2a+1$? [duplicate]

Is there a way to find all integer triplets of $(x, m, a)$ for the following equation. $$\frac{2^m-1}{2^{m+x}-3^x}=2a+1$$
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Number of possible cycles in collatz conjecture

I had been following all the blogs, but I would like to understand, whether an attempt has been made to understand how many cycles are possible apart from the 1-4-2-1 cycle in collatz problem
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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 ...
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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 ...
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35 views

Collatz sequences meet at a point

I was solving the problem,in which we are given two starting values of collatz sequence and our task is to say after how many steps their sequences “meet” for the first time. For ex - a= 7 , b= 8 a ...
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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 ...
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2answers
126 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?
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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 ...
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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 ...
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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 ...
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158 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 ...
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157 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 ...
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1answer
122 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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85 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 ...
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106 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 ...
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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 ...
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1answer
123 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 ...
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1answer
70 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, ...
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1answer
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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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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
46 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 & ...
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2answers
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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, ...
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1answer
107 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 ...
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1answer
101 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 ...
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1answer
273 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.
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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 ...
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2answers
133 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 ...
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1answer
165 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 ...
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158 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 ...