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Questions tagged [collatz]

for questions about the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even, and its generalizations to $n \to pn+q$ .

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Cycles in bit-limited verson of Collatz map

By 'bit-limited', I mean that we have a computer/calculator that handles binary numbers up to $n$ bits ($n>=1$), and any numbers greater than $2^n-1$ overflow by truncating the higher bits. ...
2
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3answers
76 views

Number of Collatz steps for Mersenne numbers

I noticed that for all $k \in \mathbb{N} \geq 1$ the following is true (I tested up to $2^{5000}$): $\text{Collatz_Steps}(2^{2k+1} - 1) + 1 = \text{Collatz_Steps}(2^{2k+2} - 1)$ Where $\text{...
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2answers
69 views

Irritating “proof” of the Collatz Conjecture

I recently stumbled across this self-proclaimed proof of the Collatz Conjecture. It seems very irritating to me that this very hard conjecture is supposedly proven by using very basic counting ...
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1answer
63 views

$3n+1$ graph and shared vertices

I found this somewhere on the web: Theorem. The total number of vertices for $n$ squares that share exactly one common vertex is given by the formula $f(n) = 3n + 1$. Proof. Each of the $n$ squares ...
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2answers
452 views

Preserving historical information of the Collatz function?

In some sense this two equations are the same, namely $f_2$ preserves the historical information of $f_1^n$, where the exponent is function composition, but I am not sure how to show this rigorously. $...
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1answer
102 views

How to prove property of Collatz Conjecture trees (3x+1 Problem)

Inevitably for any amateur mathematician, I've been playing with the Collatz Conjecture. I have found it's easier to examine and to generate theorems if we use this equivalent statement: $$c_{n+1}\...
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1answer
86 views

"What is the specific mathematical reason behind the origin of the Collatz Conjecture that makes it difficult to solve it? [closed]

Is there a known so specific mathematical reason that makes it difficult to solve the Collatz Conjecture? Clearer: What is the specific mathematical reason behind the origin of the Collatz ...
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1answer
102 views

Why is the Collatz Conjecture so difficult to prove or disprove? [duplicate]

Is there some quality of the Collatz Conjecture that has made it so difficult to prove or disprove? Besides just using a computer to calculate lots and lots of values of $n$, of course.
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2answers
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Collatz $2x + 1$ conjecture?

Do we know of any Collatz theorem involving similar functions. For example what do we know about iterations of: $$ f(x) = \begin{cases} \dfrac{x + 1}{2} \text{, if } x \text{ is odd}. \\ 2x + 1, \...
1
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1answer
134 views

Polynomial divided by a monomial in some extended Collatz Shortcut function

Quirks of the Collatz shortcut function. One example and some basic questions. The functions here are defined as: $$C_d(n):\mathbb{Z^+}\rightarrow\mathbb{Z^+}$$ $$n\in\mathbb{Z^+}$$ The definition ...
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2answers
153 views

Is there a polynomial (or series) expression for summing $S_d(a,N)=\sum_{k=0}^{N-1} \log(1+{1\over a+k \cdot d})$? (perhaps Bernoulli-type)

I need a quickly evaluatable expression for sums of consecutive logarithms of the type $$ S_{d}(a,N) = \log(1+ {1\over a})+\log(1+ {1\over a+d})+\log(1+ {1\over a+2d})+ \cdots + \log(1+ {1\over a+(N-1)...
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3answers
162 views

$k$-cycles in Collatz functions

I have a couple of questions, but I need to give some quote and some reasoning before I ask. Quote from Wikipedia: A $k$-cycle is a cycle that can be partitioned into $2k$ contiguous ...
3
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1answer
110 views

There are lower bounds worked out for the length of nontrivial Collatz-cycles. How can *upper bounds for the disproof* be determined?

There have been lower bounds estimated for the length $N$ of (odd) steps of a nontrivial cycle in the collatz-problem. Such estimates have been based on knowledge of upper bounds $\chi$ for any number ...
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2answers
496 views

What is currently the highest lower bound for the length of a nontrivial cycle in the Collatz Conjecture?

We know that there are two possibilities to disprove the Collatz Conjecture. We find a nontrivial cycle. We find a sequence that diverges to $\infty$ A non-constructive disproof is imaginable as ...
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0answers
107 views

Could the Collatz Conjecture be related to Solitaire?

When observing different variations of the Collatz Conjecture, I found that stumbling across loops instead of watching the trajectory decay to one happens a lot. When playing a game of Klondike ...
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1answer
47 views

Literature request: Papers by Charles C. Cadogan.

I am looking for three papers on the Collatz Conjecture by Charles C. Cadogan. The Annotated Bibliography lists these three as: "Charles C. Cadogan (2000), The 3x+ 1 problem: towards a solution, ...
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2answers
77 views

Does anyone know where I can get a copy of R. P. Steiner, “A theorem on the Syracuse problem”?

R.P. Steiner. "A theorem on the Syracuse problem". In: ed. by D. McCarthy and H. C. Williams. Congressus numerantium; 20. Proceedings of the 7th Manitoba Conference on Numerical Mathematics and ...
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1answer
84 views

Is this Algorithm's Result concerning Collatz Sequences provable?

I have searched the www for this little algorithm and its result thoroughly, but it was nowhere to be found, do you know if my observation is provable? The Algorithm: Step 1: Select an arbitrary ...
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0answers
40 views

Prove the following inequality (number theory) [duplicate]

If $2^m > 3^n$ for some $m>n>1$ then show that $$2^m - 2^{(m-n)} > 3^n$$ I've verified the inequality for a large number of values of $n$ and $m(n)$, where $m(n) = \lceil n\log_23\rceil$, ...
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3answers
108 views

Is there a maximum number of consecutive decreasing steps a Collatz cycle can have?

If we take a look into the (known) cycles of the Collatz Conjecture when all integers are included, we get 4 cycles: 1 → 4 → 2 → 1 … −1 → −2 → −1 … −5 → −14 → −7 → −20 → −10 → −5 … −17 → −50 → −25 ...
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Variation Of The Collatz Conjecture Discovered [closed]

Consider the following operation on an arbitrary positive integer: If the number is divisible by 12, divide it by 12. If the number is divisible by 10, divide it by 10. If the number is divisible ...
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47 views

What terminology should I use when refrencing how close a sequence is to a loop for research?

I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping. For example, If I was interested in this ...
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1answer
98 views

How is this collatz conjecture?

In this video (don't have to see the video as I'm going to give the requierd info) the author defines the $collatz(x,y)$ function as below : If $x = 3y$ or $y = 3x$ or $x = 2y+1$ or $y = 2x+1$, ...
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0answers
66 views

Which numbers will iterate to others under the Collatz iteration?

I have a question about the Collatz conjecture and how some numbers merge trajectories. Take the standard map: $$C(n) = \begin{cases} n/2, & \text{if $n \equiv 0$ mod $2$} \\ 3n+1, & \text{...
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1answer
56 views

When given a non-multiple of $3$, $k$, is it possible construct $m<k$ with these conditions? [closed]

This is another Collatz-related problem about trying to represent a number in a certain form. As is usually the case with the Collatz conjecture, this is probably not useful. My question is : Can ...
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2answers
189 views

Can any integer, not a multiple of three, be represented as $n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$?

This question has some relevance to the Collatz conjecture. It was originally based on trying to represent a number like this: Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+...
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1answer
85 views

infinite sum can't be natural

Take the following sum. $$f(k,p)=\frac{\sum_{i=1}^k2^{p_{i-1}}3^{k-i}}{2^{p_k}-3^k}$$ where the set $p$ is an arbitrarily increasing set of positive integers. Explain why $f(k,p)$ is only natural when ...
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2answers
115 views

For each $\epsilon$, for which $\delta$ does $d(x,y)<\delta\implies d(f(x),f(y))<\epsilon$ hold?

I have the definition : A function from a metric space to a metric space is uniformly continuous if for all $\epsilon>0$ there exists $\delta$ such that $d(x,y)<\delta\implies d(f(x),f(y))<\...
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0answers
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How do I define an appropriate diameter operator?

I've distilled the Collatz conjecture to this, and I need to understand the next step which is to think about how to correctly define a $\operatorname{diam}$ operator with the desired properties. I ...
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1answer
179 views

Particular base-10 digit patterns in Collatz

Note: I will use the abbreviation RCF for the Reduced Collatz Function. The arrangement of certain specific digits produce a particular pattern on the next iteration of the Reduced Collatz Function. ...
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1answer
103 views

What's the closure of the equivalence classes of numbers the same distance from $\langle2\rangle$ by the Collatz graph?

What's the closure of the equivalence classes of numbers the same distance from $\langle2\rangle$ by the Collatz graph? Let the Collatz function be $f(x)=3x+2^{\nu_2(x)}$ The Collatz conjecture ...
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1answer
133 views

The Periodic Collatz Conjecture

Consider the function $$f(n)=\begin{cases}n/2&\mbox{if }n\mbox{ is even}\\3n+1&\mbox{otherwise}\end{cases}.$$ Starting from any positive integer $x_0$, we can iterate the sequence $x_1=f(x_0)...
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1answer
141 views

Collatz “factorization”

The collatz conjecture states that every number eventually reaches $1$ under the repeated iteration of $$ f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \...
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1answer
113 views

What are the elements of $\Bbb Z[\frac16]/{\sim}$ and what do the subgroups and orders of elements look like?

What are the elements of $\Bbb Z[\frac16]/{\sim}$ and what do the subgroups and orders of elements look like? In which $\exists i\in \Bbb Z:4^ia=b\implies a\sim b$ What's the identity element, for ...
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1answer
512 views

What number has 62,118 steps ? (Collatz conjecture)

According to this site : https://www.nitrxgen.net/collatz The maximum steps registered so far for a given number (with less than 500 digits) is 62,118, do we know what number it is? I tried to ...
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1answer
144 views

Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?

Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research. There do not exist $a,b$ and $c$ such that$$ (2a-1)(2^{(b+c)}-3^c )=2^b-1 $$where $a>0,b&...
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2answers
116 views

Find a formula for the collatz branch numbers

Context: Collatz conjecture What I call a 'branch number', is a number accessible by 2 different routes. Example : 24 is not a branch number, it can be accessed only from 48 (division by 2) 16 is a ...
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101 views

Similarities of CA Rule 150 and Odd Collatz-function outputs

I made a "discovery" a couple of weeks ago in regards to the first iterates of (odd) numbers of the form $2^n-1$ where $n\in\mathbb{N}$. First iterates is a bit loose term here; what I mean is all of ...
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2answers
145 views

Smallest element of cycle of length $k$ in Collatz 3x+1 map?

In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $...
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3answers
282 views

Can anything be proven about this complex variant of the Collatz problem, or is it just as intractable?

Given a Gaussian integer $z = a + bi$, where $a, b \in \mathbb{Z}$, $i = \sqrt{-1}$, iterate the function $$f(z) = \frac{z}{1 + i}$$ if $z$ has even Gaussian norm (that is, both $a$ and $b$ are odd, ...
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1answer
138 views

Is it possible to narrow down a domain of possible counter-examples to the Collatz Conjecture?

First of all, I am not trying to prove the Collatz Conjecture. I want to know if it is possible to rule out certain values of a counter-example. Suppose $k \in \Bbb Z^+$ is the lowest counter-...
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3answers
119 views

Explain the odd/even inequality in the heights of numbers under the Collatz $(3x+1)/2$ transformation?

My kid asked me a question and I'm finding it hard to answer: if every number under repeated application of the Collatz transformation1 eventually reaches $1$, then it must be true for both even and ...
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1answer
137 views

Collatz 3n+1 problem [closed]

Does it suffice to say that, using the tree/reverse of 3n+1 problem, if we can show that we can generate all positive integers starting from integer 1, including 1 itself, is equivalent to proving ...
4
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1answer
128 views

Contributing to a unsolved problem and writing a paper about it

How do I write a paper on the Collatz Conjecture without fooling or making an ass out of myself? Im not affiliated to any University at the moment (my past Uni was like 20 years ago, but I didnt ...
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175 views

Cellular Automata on the Collatz Conjecture

I have a Cellular Automaton that of any initial integer (initial condition of the automaton) generates states of Collatz sequences. The neighbourhood of the automaton is shaped like an L-tetromino (...
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141 views

Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+t_1}3^{n-3} … + 2^{t_0+t_1+…+t_{n-1}})}{3^n}$ can describe all integers

I was working on the Collatz conjecture(while not expecting to get anywhere) and I suspect that if it is possible to show that $$\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+t_1}3^{n-3} .... +...
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1answer
198 views

The orbit of $2^n+1$

Consider the Collatz function, $$T(n)=\frac {n}{2}, \text { if $n$ is even}$$ and $$ T(n)=\frac {3n+1}{2}, \text { if $n$ is odd}$$ Observing the orbit of $2^n+1$, I came up with the following ...
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1answer
119 views

In the $mx+1$-problem: headache with a lower bound for the minimal element of a cycle …

I'm trying to optimize my search-routine for cycles in the $mx+1$-problem and seem to have a knot in my brain trying to make sure a certain lower bound for the minimal member $a_1$ is reliable. ...
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0answers
259 views

What is known about the $7x+1$ problem?

One of the most famous problems in mathematics that remains unsolved is the Collatz conjecture. I am concerned with similar 7x+1 problem. I have already seen this problem mentioned in the literature....
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1answer
163 views

Convergence of iterated functions like Collatz in a Compact metric space

We have seen that functions like the Collatz converge to the same cycles: $2\rightarrow1$ no matter what inputs we have tried feed into that function. But we have yet no proof it can reach some other ...