Questions tagged [collatz-conjecture]

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.

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1answer
82 views

Collatz-esque dynamical problem about prime distribution

I've come up with a scenario which reminds me of the Collatz conjecture in that it's a question about the behavior of a system over time. Let $n=0$ at $t_0$ (i.e. $t=0$). $t$ will increment through ...
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32 views

Is $q\cdot\lvert\cdot\rvert$ a norm?

Let $\lvert x\rvert$ be a ($l^\infty$)norm on a sequence space. Is $q\cdot\lvert x\rvert$ also norm for some positive rational number $q$? This is the motivation for the question: It turns out ...
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1answer
60 views

prove only one positive whole number solution (for every integer $y$)

I have created an equation that might be able to solve half of the Collatz conjecture, but it requires a proof that is beyond me to make, and so I ask you. for all positive integers of $y$ $$x = \...
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1answer
40 views

Collatz extensions

Because, in the basic Collatz algorithm, odds are always transformed into evens (via x=3x+1), and evens are transformed into evens with a probability of .5 (via x=x/2), there will be, on average, 2 ...
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1answer
145 views

Collatz function iterates as a Markov chains

In the Collatz Conjecture, the question is wether the sequence of iterates will always reach $1$ for all $n>0$ where $n\in\mathbb{Z^+}$. The formal Collatz function consists of the operations: $n/2$...
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2answers
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Is this a valid proof of the Collatz theorem? [closed]

Is this a valid proof of the Collatz Conjecture? For all even numbers x >= 20, if all numbers 1 through x are in Collatz set, show x+1 and x+2 are in set. Then by induction, all positive integers ...
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1answer
74 views

A challenging problem on sequences. [closed]

Define $f:\mathbb N \to \mathbb N$ by $f(n)=3n+1$ if $n$ is odd $f(n)=n/2$ if $n$ is even.Now can we show that for every $x$ in $\mathbb N$ there is $k$ in $\mathbb N$ such that $f^k(x)=1$ where $f^r=...
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Another amateur attempt at the infamous Collatz Conjecture

Here is a better proof. For odd number n, either n=4a+1 or n=4a+3 for some integer a. If [n] mod 4=[1], then C(4a+1)=3*4a+4. C(C(3*4a+4))=3a+1<4a+3. As C(C(n)) If [n] mod 4=[3], n=4a+3. If 2^n-1 ...
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How many Collatz conjecture elimination patterns are there?

The Collatz conjecture is an amazing thing. For people who don't know, the Collatz conjecture is the conjecture where you take any natural number. If that number is even, divide by two. If it is odd, ...
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1answer
43 views

Are there specific total stopping times that are finite for the collatz conjecture? [closed]

For example, are there only certain groups of numbers that have a total stopping time of 6 or 30?
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135 views

Are there any known Collatz-like results?

The usual argument in favor of the Collatz conjecture (or at least in favor of there being no unbounded trajectories) essentially argues that, if we have the "shortcut" function defined by: $$f(2x)=x$$...
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2answers
241 views

Updated: Can someone have a look at a simple attempt towards a direct mathematical proof of the Collatz conjecture?

I am looking for feedback on the below, any is appreciated :) Remark: Apologies in advance for any incorrect use of notation as my mathematical experience is quite novice, additionally a word of ...
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1answer
106 views

Rework this statement to the case $c=\frac13$ and $\sim=\langle2\rangle$

This comment reads: Let $g(x) = x 2^{-v_2(x)}3^{-v_3(x)}$ and $F(x)=x+c$ or any function then $f(x)=g(F(g(x))$ can be seen as a function $\Bbb{Q^*/<2,3> \to Q^*/<2,3>}$. There are no ...
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Finding the stopping times or steps for positive integers in the Collatz conjecture using a formula

Is it possible to find a closed-form $a_n$ for some positive integers and use another formula to find the number of steps for all of those integers? If so, can you find multiple closed forms? For ...
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153 views

Explicit results for the Collatz Conjecture? [closed]

I have some short questions I'd like to ask. I hope I'm asking in the right place. Has there been any significant progress on the Collatz Conjecture to date? What Tao really did was to set ...
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62 views

Collatz sequence vs A180076 (OEIS)

I found a sequence in OEIS (A180076) where a subsection is similar (but not entirely) to trajectory of $27$ in Collatz function. In the comments section I read: "Permutation of the natural numbers ...
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79 views

Can Collatz conjecture be formulated as totality problem for a 2-tag system?

I have just read about Terence Tao's approaches to Collatz conjecture, and I have a stupid question. If binary representations of numbers in the hailstone sequence for Collatz conjecture can be ...
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92 views

Is the inverse Collatz graph symmetric?

Maybe a stupid question, but I couldn't find a direct answer. When looking at this beautiful visualization of the graph of inverse Collatz sequence: https://www.jasondavies.com/collatz-graph/Collatz ...
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1answer
77 views

Can the uniqueness of the solution be proved?

Let $$t = \frac{3^4 + 2^k(3^3 + 2^l(3^2 + 2^m(3^1 + 2^n)))}{2^{p + n + m + l + k} - 3^5}$$ where k, l, m, n, p can only be natural numbers >=1. The unknown t has also to be a natural number >= 1. ...
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1answer
90 views

What's wrong with this reasoning concerning collatz cycles?

Say there exists a collatz cycle: $k_0, k_1...k_{m-1}, k_0$ This means that: $$3^a k_n + \sum^{a-1}_{i=0} 3^i*2^{b_i} = 2^c k_n $$ where $b_i > b_{i+1}$ and $b_{a-1} = 0$ and since there are no ...
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1answer
60 views

Collatz Cycles with double the even numbers than odd

This question is with reference to the collatz conjecture. It is known that a number $A$ in a collatz cycle is of the form: $$A = \sum_{i=1}^k \frac{ 3^{k-i} \cdot 2^{k_i}}{2^n - 3^k},$$ $0 = k_1< ...
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1answer
930 views

Calculate the maximum in the Collatz sequence

Consider the notorious Collatz function $$ T(n) = \begin{cases}(3n+1)/2&\text{ if $n$ is odd,}\\n/2&\text{ if $n$ is even.}\end{cases} $$ One of the most important acceleration techniques of ...
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224 views

What is the flaw in this proof of the Collatz Conjecture?

Would love any feedback for this proposed proof of the Collatz Conjecture. (For more details explaining each step, I made a video here: https://www.youtube.com/watch?v=P0F4zbNdbTU ) Ignoring all ...
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1answer
94 views

What is the finest topology that makes $\lim_{n\to\infty} x+(1-2^{-6n})\cdot2^{\nu_2(x)}\cdot3^{\nu_3(x)-1}$ continuous?

Let $M=\Bbb Z[\frac16]^+$ be the multiplicative monoid of positive dyadic and ternary rationals. Let $Q=\Bbb Z[\frac16]^+/\langle2,3\rangle$ be the quotient that sets $x\sim y\iff\exists p,q\in\Bbb Z:...
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2answers
130 views

How fast are current programs computationally verifying the Collatz conjecture?

I have seen several past or ongoing projects trying to computationally verify the Collatz conjecture. These projects can be divided into two groups according to the algorithm they use: The first ...
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1answer
184 views

Different approach to solving the Collatz problem

Edit: A paper by Joseph Sinyor can be found here, and has a small section on The 3x+1 Problem and Mersenne Numbers, I think it is somehow relevant to what I was trying to deliver here. The ...
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92 views

Collatz Conjecture Numbers up to n

I was playing around with the Collatz sequences of numbers up to a number. My question was, for which numbers $n$ does no integer below $n$ reach $n$ in its iteration? So I wrote a program to find ...
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1answer
64 views

Is $\lvert 3x+2^{\nu_2(x)}\rvert\leq\lvert x\rvert$ in the monoid quotient $\Bbb N^+/\langle3,4\rangle$?

What does the monoid quotient $\Bbb N^+/\langle3,4\rangle$ look like, where $\Bbb N^+$ is the multiplicative monoid produced by multiplying primes? In particular, is it by any chance relatively quick ...
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194 views

Proof of $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m$

How would I go about proving the following: For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that, $...
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1answer
73 views

What is the topological entropy of the Collatz map (extended to 2-adic integers)?

In the process of learning the basics of dynamical systems, I finished the chapter on topological entropy and decided, as an exercise, to try and compute the entropy of one of my favorite maps: $T : \...
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1answer
121 views

3n+1 problem: $f^x(n) < n$ finitely many steps

Consider the shortcut function: $$f(n)=\begin{cases}(3n+1)/2\} & \text{if } (n\bmod 2)\equiv1 \\ \{n/2\} & \text{if } (n\bmod 2)\equiv0 \end{cases}$$ Define $f^x$ as the $x$'th iterate of $f$....
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1answer
55 views

Natural functions that converge to 1

What are some examples of functions that given a positive integer x converge to 1 using only natural operations (add, sub, div, mul). An example would be the collatz conjecture, although it requires ...
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1answer
154 views

Ratio of (Collatz “path” bits) / (starting number bits) $\approx 2$

The "Path Records" page by Eric Roosendaal related to the Collatz conjecture (see here), reports a "striking" feature of the path records. If we define $\text{MaxSeq}(M)$ (in Roosendaal it is $\text{...
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1answer
62 views

How do I correctly (or better) define this bound or constraint in the padic ring $\Bbb Z_2$?

Let $T(x)$ be a partial function $:\mathcal P(\Bbb Z^+)\to\Bbb Z_2$. I have the rule for a certain $\underline x\in \mathcal P(\Bbb Z^+)$ that at least every fourth digit of $T(\underline x)$ is a $1$...
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2answers
168 views

Do I understand the concept of $x^{0.84}$ correctly?

I'm trying to understand the concept of $x^{0.84}$ that Jeffrey Lagarias found for Collatz Conjecture. If I'm wrong, please correct me with an answer. I understand such that, Suppose the interval $\...
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0answers
94 views

Distributed computing project to check the convergence of the Collatz problem

I am about to start a distributed computing project to check the convergence of the Collatz problem. So far, the convergence of the problem has been verified for all numbers below $87 \times 2^{60} \...
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1answer
157 views

Is there a Collatz 'branch' stopping time equation?

Definitions: -Collatz: odd: $\frac{3x+1}{2}$ | even: $\frac{x}{2}$ -Branch: Starting at an odd number and increasing until reaching an even number. Then decreasing until reaching another odd number (...
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3answers
152 views

The $k^{th}$ iteration of $2^k-1$,$2^k-3$ ,and $2^{k}-11$ and others

With the short cut version of the $3x+1$ conjecture, $$f(x)=\frac {3x+1}{2},\text {if $x$ is odd }$$ and $$f(x)=\frac {x}{2},\text {if $x$ is even }$$ numerical experiments show that $$ f^k(2^k-1)=3^{...
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0answers
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The periodic orbits in $3x+1$ conjecture on $\mathbb {Z}$ [duplicate]

The $3x+1$ conjecture is defined on $\mathbb{N}$ as $$ f(x) = (3x+1)/2 $$ if $x$ is odd and $$f(x)=x/2$$ if $x$ is even. As far as I know the only periodic orbit is the trivial $$1\to 2 \to 1\to 2.....
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314 views

Showing $3^iq-2^{i-p}\neq2^pq-1$, with $p:=\lceil i\,\log_23\rceil$, $q:=\left\{\frac{2^{i}\,3^{2^{p-i-2}}}{2^p3^i}\right\}$, $i>6$

For this inequality $$3^i\cdot\left\{\frac{2^i\cdot3^{2^{\lceil i\cdot \log_23\rceil-i-2}}}{2^{\lceil i\cdot \log_23\rceil}3^i}\right\}-\frac{2^i}{2^{\lceil i\cdot \log_23\rceil}}\leqslant 2^{\lceil i\...
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3answers
292 views

Concept of undecidability and the Collatz problem [duplicate]

I'm reasoning like this: (I don't have enough math education) Suppose that the Collatz Conjecture is undecidable. This means, the Collatz conjecture cannot be proved to be true or false. This means, ...
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2answers
151 views

Why problem with simple formulation is so hard?

If you ever heard about Collatz conjecture, you know that it is understandable even for middle school students, but no one has solved it yet. The problem is to prove or to disprove that starting with ...
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1answer
398 views

Application of the Collatz conjecture

I'm very curious about the Collatz conjecture, also known as the $3n+1$ problem, mainly due to its rather simple formulation and beautiful visualizations. After all, Erdős himself said that "...
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Base-10 “reverse pairs” in strings of the iterated Collatz conjecture function.

Details: The following terminology is non-standard. Definition 1: A reverse pair of numbers is a pair of numbers $m$ and $n$ such that, if the decimal expansion of $m$ is $$\overline{a_1a_2\cdots ...
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3answers
230 views

Infinite sets that are proven to be true for Collatz Conjecture?

I know that the set of powers of 2 and the set of 1,5,21,85,341,... are proven to be true for Collatz conjecture. Are there other sets with infinite number of numbers that are also proven to satisfy ...
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1answer
816 views

Computational verification of Collatz problem

Every positive integer $n$ can be represented as a product \begin{align} n &= a \cdot 2^k \text{,} \end{align} where the $a$ is odd integer and $k$ is an exponent of two. Let $\varepsilon$ be the ...
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1answer
136 views

Is it officially proven that the Collatz sequences can't go to infinity?

Did the mathematician Jeffrey Lagarias prove that in his work the Collatz sequences could not go to infinity (divergent trajectory), that only cyclicity can exist? I don't have enough mathematics to ...
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1answer
84 views

On a one formula Collatz function

The top voted answer in Is it possible to describe the Collatz function in one formula? looks like a really complicated function at first sight. Let me restate the function in it's original form here;...
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1answer
874 views

How far has Collatz conjecture been computationally verified?

This page from 2017 by Eric Roosendaal says that the yoyo@home project checked for convergence all numbers up to approx. 266. Is it still a valid record? I am aware of the ongoing BOINC project, but I ...
2
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1answer
351 views

Alternative formulation of the Collatz problem

Just for the sake of interest, I have realized that the additive step in the Collatz function can be technically avoided when computing the function iterates. Rather than defining the Collatz ...

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