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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27 views

Determinant of matrix transpose and Laplace expansion, application

I recently read this paper in which the authors construct a matrix related to the Collatz conjecture such that $$ m_{ij} = \begin{cases} 1 \text{ if } i = j\\ x \text{ if } c(i) = j\\ 0 \...
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79 views

Erdös said that math is not ready for the $3x+1$ problem. What is the reasoning for that?

I know the Collatz conjecture is dangerous ground and laymen should not be playing here. Please bear with me. I'm interpreting Erdös' sentence as, we are lacking tools to tackle such kind of problems....
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2answers
94 views

A conjecture related to collatz

I will assume that the reader knows the Collatz (3n+1) conjecture. Terminology: let's say that a natural number $ n $ is a descendant of $ m $ if the collatz procedure starting at $ m $ eventually ...
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46 views

The length of the algorithm [closed]

The "3n + 1 algorithm" works as follows. Start with any number n. If n is even, divide it by 2. If n is odd, replace it with 3n + 1 So, for example, if we start with 5, we get the list of numbers 5,...
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1answer
52 views

induction for the collatz conjecture for $n=2^k$

Prove / disprove the following statements regarding the Collatz conjecture T $(1) \forall n \in \mathbb{N} ((\exists k \in N_{0} \hspace{0.5cm} n=2^k ) \rightarrow T(n)=1)$ $(1) \forall n \in \...
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1answer
67 views

Generalized Collatz problem mx+r [closed]

I'm looking for scientific papers on $9x+r$ and $11x+r$ problem. I once read a paper in which it was stated that no cycles were found in these sequences and probably all sequences are divergent to ...
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102 views

Can Anyone Explain This Property of the Collatz Conjecture ($3n + 1$ problem)

I'm making a program that draws out a tree containing the first $n$ numbers, and how they all reduce to $1$. I noticed something interesting about the number of nodes that are required to connect ...
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63 views

A first step into solving the Collatz Conjecture — a generalization

I have tried working with an infinite iterating equation, that may help in solving the Collatz conjecture (3n + 1) / 2 Step one: pick any number, so let's say, for this question, 2 Step two: choose ...
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494 views

Binary representation of Collatz numbers — what is known so far about the maximum number of divisions by $2$?

Based on the binary representation of Collatz numbers, we proved that the maximum possible number of divions by two, $\hat\alpha$, in a Collatz sequence is given by $$\hat\alpha=\lfloor n\cdot\...
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1answer
131 views

Transform Collatz sequence to a strictly decreasing sequence

While playing with numbers, I found that every Collatz sequence $n, T(n), T^2(n), \ldots, 1$ can be associated with a strictly decreasing sequence of integers. The Collatz conjecture asserts that a ...
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Collatz sequence lengths for Primorials

The primorial $(\#)$ function is defined as the product of first $n$ prime numbers. Is there a simple explanation for the observation that the length of the Collatz sequence for successive primorials ...
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1answer
44 views

Let $x=2^{p-1}-1$ be composite, $p$ prime and $3\mid x$. Why $p \mid x$?

Let $x=2^{p-1}-1$ be a composite odd natural number (a "wrong" Mersenne) and $p$ is prime and $3\mid x$. Why $p \mid x$? Does really $p$ always divides $x$? Note: We know that $x$ is a sum of powers ...
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Is it possible to prove this conjecture about Collatz sequences?

I have constructed a tree that looks something like this: $$ 1,2,4,8,16, \begin{cases} 32,64,\begin{cases} 128, 256, \begin{cases} 512...\\ 85... \end{cases}\\ 21, 42, 84... \end{cases}\\ 5, 10, \...
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1answer
94 views

$3n+1$ Collatz generating Chen Primes

I Ask it just for fun: Consider the Collatz sequence $3n+1$. $x$ is a Chen-prime-$3x+1$ record holder, if for all $n<x$, the $3n+1$ sequence produces less Chen-primes than $3x+1$ before reaching $...
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1answer
171 views

These equations can generate integers that have the same total stopping time in the Collatz Conjecture. Has this been discovered?

Conjecture #1: $a_(x,y)=(1/3)(2^{(2y+7)-2x)}(5(4^{x})-2)$ Generates positive even integers with a total stopping time S where $S=2y+13$, y is the set of all natural numbers and x is the set of all ...
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134 views

Is the sum $\sum_{i=1}^{n}\frac{2^{\alpha_1+\ldots+\alpha_i}}{3^iv_1+\sum_{j=1}^{i}3^{j-1}2^{\alpha_1+\ldots+\alpha_n-\sum_{l>i-j}\alpha_l}}$ limited?

Does the sum $\sum_{i=1}^{n}\frac{2^{\alpha_1+\ldots+\alpha_i}}{3^iv_1+\sum_{j=1}^{i}3^{j-1}2^{\alpha_1+\ldots+\alpha_n-\sum_{l>i-j}\alpha_l}}$ for a positive odd integer $v_1$ and a sequence of ...
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1answer
114 views

Does my definition for “compressive functions”/“compressor recursions” make sense?

I'm trying to work on a theory of "compressive functions"/"compressor recursions"—functions which, when applied recursively to a domain (which I'll probably restrict to being discrete) bring all ...
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2answers
411 views

Is this proof for no non-trivial Collatz cycles flawed?

I've been working with the Collatz Conjecture a lot lately as a way to distract me from the math I'm supposed to be doing for school. Anyway I feel like I've constructed (not very rigorously I might ...
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Multiplication of permutations and generalization of the Collatz sequences - a difficult problem

I have a problem that seems to be very difficult. It's about multiplying some specific permutations. Let's take recursive functions that are a generalization of the Collatz sequences: $\...
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Is it plausible that $\overline{\langle3\rangle}$ being a rank two subgroup of $\Bbb Z_2^\times$ implies this graph comprises two connected subgraphs?

Let $P_1$ be the (known true) predicate: the 2-adic completion of $3^m$ is a rank two subgroup of $\Bbb Z_2^\times$ the multiplicative subgroup of the $2$-adic ring of integers. Let $P_2$ be the ...
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3answers
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Does 16 always show up in Collatz Conjecture number leading down to 1?

Just wondering if 16 always shows up in number > 16 manipulation process leading down to 1 through Collatz conjecture? or are there any exceptions? I happen to see this link with the animation ...
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1answer
102 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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1answer
81 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
57 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
176 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
82 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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69 views

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
58 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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151 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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1answer
274 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
111 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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101 views

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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164 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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70 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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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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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
78 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
91 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
64 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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2answers
1k 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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272 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
104 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
163 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
201 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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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
65 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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200 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
78 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
127 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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