# Questions tagged [collatz-conjecture]

The Collatz Conjecture, also known as the 3n+1 conjecture, is a famous open problem named after Lothar Collatz.

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### 7 for 5n+1 series diverges to infinity? [closed]

Write an odd integer in the "modified" binary form as $2^m - 1$ and apply 5n+1 (similar to applying the rules of original 3n + 1) to it:
\begin{align}
n &= 2^m - 1 \\
...

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### Cocycles and the Collatz problem?

Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...

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### Finding a two point scrambled set for the function $g:[0,1] \rightarrow [0,1], x \mapsto \min_{n\in \mathbb{Z}} |3x-2n|$?

Let $I=[0,1]$ be the unit interval and $g$ as defined below.
Then $x \neq y$ with $x,y \in I$ are called "two point scrambled set"=$\{x,y\}$, if
$\lim\inf_{n \rightarrow \infty} | g^{(n)}(x)...

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### Is it known that the the sequence 7n+1 diverges to infinity starting with 7?

In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...

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### Can the Collatz conjecture be independent of ZFC? [closed]

It is known that the Continuum Hypothesis is independent of ZFC.
The formulation of the Collatz conjecture looks somehow more simple than that of the Continuum Hypothesis.
Is it possible that the ...

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### A question and reference about Bombieri's article continued fraction of algebraic numbers

Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...

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### Assuming the Collatz conjecture is false, what is known about the size of the false set?

If the Collatz conjecture is strongly false, in the sense that there is an infinite orbit, let $S_n$ be the set of natural numbers $\le n$ whose orbit goes to infinity.
If $c=\liminf _{n\rightarrow\...

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### A problem similar to the $3x+1$-problem [closed]

Let $n$ be a fixed positive integer. Define the function $f_n(x)$ as follows:
$$f_n(x)=\left\{\begin{aligned}&2x-1,\quad x\leq n;\\&2(x-n),\quad x> n.\end{aligned}\right.$$
and for $l\in\...

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### Extended Collatz conjecture

As you all know, the Collatz conjecture claims that any positive integer will eventrually be reduced to 1 by appllying the sequence $n_{i+1} = x*n_{i} + 1$, when $n_{i}$ is odd, and $n_{i+1} = n_{i} / ...

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### Finding a strictly increasing Collatz sequence of arbitrary length [closed]

Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I ...

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### A mutation of the Collatz disease

Given $k \in \mathbb N$, we define $f_k: \mathbb N \longrightarrow \mathbb N$ by
$$ f_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } ...

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### First odd term of the sequence lower odd number $n$ related to the $3\cdot n+1$ problem

I have already asked on math.stackexchange if you think the question is off topic I can delete it.
I'm trying to complete the following graph but I'm not sure if I can complete it without getting an ...

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### How would one go about solving this conjecture concerning exponential Diophantine equations?

I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...

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### If proven that all Collatz sequences attain bounded values, is it also proven that all sequences end up below the number you start from?

I was researching upon the Collatz conjecture, and I was reading all the research work done by mathematicians including Terry Tao's. I had read that before Terry Tao's research it was proven that ...

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### Reference on the Collatz conjecture [closed]

I'm just looking for references in the literature for some observations I made for fun about the Collatz conjecture.
The Collatz conjecture states that any positive integer $n$ can eventually be ...

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### The two Collatz-maps associated to characters modulo 8

Given a Dirichlet character $\chi$ modulo $8$ we consider the map $\mu(x)=x/2$ if $x$ is even and $\mu(x)=(3x+\chi(x))/2$ otherwise.
(The corresponding map for $\chi$ the trivial Dirichlet character ...

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### Has the Collatz been investigated as a recursive function?

Does anyone ever write the Collatz conjecture as a single algebraic, recursive sequence? For example, a crude version might be:
$$
g(n+1)=\delta _{1,g(n)}+(1-\delta _{1,g(n)})*\left(\left(\frac{cos(\...

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### Can anyone recommend a reference where the collatz conjecture is viewed as a combinatorics problem?

It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ...

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### Polynomials, $3^x$ and the Collatz conjecture

$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open.
Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...

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### Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?

The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...

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### Can you explain this weird pattern in Collatz conjecture? [closed]

Extreme disproportion in the dispersion of "Digital Roots" of the highest numbers reach in the Collatz Conjecture.
I calculated the Digital Root remainder mod 9 for the highest numbers ...

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### Summary of “Almost All Orbits of the Collatz Map Attain Almost Bounded Values”

Terence Tao's 2019 paper ``Almost all Orbits of the Collatz map attain almost bounded values" is pretty famous. However, it's also long and complicated. I think there are useful techniques to ...

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### Does this iterating process always returns to 0 for positive $a_0$?

Given $a_0$ be an positive integer, define
$$ a_{n+1} =
\begin{cases}
8a_n, & \text{if $a_n$ is odd} \\
\lfloor a_n/3\rfloor, & \text{if $a_n$ is even}
\end{cases}$$
Now form the sequence $(...

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### Surreal numbers and the Collatz iteration as a game?

Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...

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### Generating functions of Collatz iterates?

Let $C(n) = n/2$ if $n$ is even and $3n+1$ otherwise be the Collatz function.
We look at the generating function $f_n(x) = \sum_{k=0}^\infty C^{(k)}(n)x^k$ of the iterates of the Collatz function.
The ...

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### Collatz conjecture and a diophantine equation

Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...

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### When is $\{b^2 - \{b-1\}_2\}_2=1$ with odd $b$? (The bracket-notation explained below)

For the complete extraction of the factor $p$ and its powers from a natural number $n$
let's define the notation $$ \{n\}_p := { n \over p^{\nu_p(n)}} \tag 1$$
$ \qquad \qquad $ Here $\nu_p(n)$ means ...

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### Generalized Collatz sequences

Let $\mathbb{N}$ denote the set of positive integers. For $k\in\mathbb{N}$ let $c_k:\mathbb{N}\to\mathbb{N}$ be defined by $x\mapsto x/2$ for $x$ even and $x\mapsto kx+1$ otherwise. The Collatz ...

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### What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance?

Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from ...

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### Collatz conjecture in all its variants

There are all kinds of execution variants to the collatz conjecture for when hitting an odd number:
$3n+1$ or $3n+3^a$ or $1.5n + 0.5$ or $1.5n + 1.5$... . The assumption is: proving any of them will ...

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### Does this prove Collatz is a $\Sigma_1$ problem?

So I got an email from one of my colleagues on the Collatz Conjecture with a link to the article Computer Scientists Attempt to Corner the Collatz Conjecture by Kevin Hartnett in Quanta Magazine.
On ...

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### Heuristic for a density conjecture related to the Collatz $(3x+1)$-problem

First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as:
$$
T(n) = \left\{ \begin{array}{}
3n+1, & \text{if $n$ is odd}\ \\
n/2, & \text{if $n$ is even}
\end{array} \right.
$$
...

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### A possibly easy question about latent geometry in Collatz sequences

I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be aware ...

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### The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd

This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here.
We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...

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### Unexpected behavior involving √2 and parity

This post makes a focus on a very specific part of that long post. Consider the following map:
$$f: n \mapsto \left\{
\begin{array}{ll}
\left \lfloor{n/\sqrt{2}} \right \rfloor & \...

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### Borderline Collatz-like problems

The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...

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### Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...

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

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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### Explicit bounds from Tao's result on Collatz conjecture

A new preprint by Terry Tao has recently appeared and has established some interesting results regarding the topic of Collatz conjecture. I will not cite the precise result, but rather an equivalent ...

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### A complementary of the Collatz $3x+1$ problem

Let $\mathbb{N}_{\text{odd}}$ be the set of odd positive integers. For $x_0 \in \mathbb{N}_{\text{odd}}$
consider the set-valued sequence $\{A_n\}_{n=0}^{\infty}$ defined by the formula
$$
A_0 = \{...

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### Two reasons why the Collatz conjecture could fail

Let $\mathbb{N}$ denote the set of positive integers. The Collatz function $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we ...

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### For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)

Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...

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### Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture.
Goldbach's conjecture asserts that every ...

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### 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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### Density of the Klarner-Rado Sequence

Consider the Klarner-Rado sequence OEIS A005658 defined by the rule: the sequence starts with 1, and if it contains $n$ it also contains $2n$, $3n+2$ and $6n+3$. According to R. Guy's popular article,...

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### How can I catalog these generalized Collatz problems?

The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$:
$x + x + 1 \rightarrow x+x+x+1+1;$
$x + x \rightarrow x;$
Whenever a number matches the LHS ...

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### Extension of Coburn's theorem on isometry and Toeplitz algebra

$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...

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### A problem involving the inverse Collatz map

Let $C$ be the Collatz map on the natural numbers, defined by:
$$C(n) :=
\begin{cases}
n/2 & \text{if} \;n \;\text{even} \\
(3n+1)/2 & \text{if} \;n \;\text{odd}
\end{cases}$$
The inverse ...

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### Identification of Invariant Sets for Discrete Dynamical Systems on the Positive Integers

Let $\phi:\mathbb{N}\times \mathbb{N}^+\rightarrow \mathbb{N}^+$ be a dynamical system on the positive integers. Suppose we refer to the orbit of a periodic point of $\phi$ as an invariant set of the ...

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### Undecidable easy arithmetical statement

Is there a basic arithmetic statement which is known to be undecidable ?
By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some ...