Skip to main content

Questions tagged [collatz-conjecture]

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

Filter by
Sorted by
Tagged with
-4 votes
0 answers
117 views

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 \\ ...
Gaurav Goyal's user avatar
0 votes
0 answers
563 views

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 ...
mathoverflowUser's user avatar
2 votes
1 answer
159 views

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)...
mathoverflowUser's user avatar
17 votes
2 answers
4k views

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 ...
pie's user avatar
  • 411
1 vote
0 answers
155 views

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 ...
Riemann's user avatar
  • 682
1 vote
0 answers
152 views

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, ...
XL _At_Here_There's user avatar
18 votes
1 answer
3k views

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\...
Yaakov Baruch's user avatar
2 votes
1 answer
263 views

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\...
Ren Guan's user avatar
  • 111
2 votes
0 answers
360 views

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} / ...
Ivan Borisyuk's user avatar
0 votes
1 answer
234 views

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 ...
NotAGhost's user avatar
3 votes
2 answers
591 views

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 } ...
Roland Bacher's user avatar
1 vote
0 answers
283 views

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 ...
user140242's user avatar
1 vote
0 answers
320 views

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 ...
John Eaton's user avatar
0 votes
1 answer
321 views

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 ...
Areen Rath's user avatar
2 votes
1 answer
1k views

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 ...
user140242's user avatar
5 votes
0 answers
168 views

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 ...
Roland Bacher's user avatar
1 vote
0 answers
416 views

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(\...
John Eddy's user avatar
-1 votes
1 answer
485 views

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 ...
Larry Freeman's user avatar
1 vote
1 answer
585 views

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 ...
Turbo's user avatar
  • 13.8k
8 votes
0 answers
1k views

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 $3n+1$...
Turbo's user avatar
  • 13.8k
-1 votes
1 answer
683 views

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 ...
Sir.Otonin's user avatar
4 votes
1 answer
2k views

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 ...
user144527's user avatar
1 vote
0 answers
396 views

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 $(...
Pruthviraj's user avatar
2 votes
0 answers
273 views

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 ...
mathoverflowUser's user avatar
11 votes
1 answer
1k views

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 ...
mathoverflowUser's user avatar
4 votes
0 answers
493 views

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 ...
mathoverflowUser's user avatar
6 votes
4 answers
485 views

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 ...
Gottfried Helms's user avatar
6 votes
1 answer
1k views

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 ...
Dominic van der Zypen's user avatar
1 vote
1 answer
502 views

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 ...
it's a hire car baby's user avatar
1 vote
0 answers
2k views

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 ...
EMN's user avatar
  • 39
5 votes
1 answer
982 views

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 ...
Turbo's user avatar
  • 13.8k
5 votes
0 answers
330 views

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

First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as: $$ T(n) = \left\{ \begin{array}{} 3n+1, & \text{if $n$ is odd}\ \\ n/2, & \text{if $n$ is even} \end{array} \right. $$ ...
mhum's user avatar
  • 1,645
4 votes
2 answers
363 views

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 ...
user918212's user avatar
  • 1,085
4 votes
0 answers
440 views

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 $n$ ...
Sebastien Palcoux's user avatar
25 votes
3 answers
2k views

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 & \...
Sebastien Palcoux's user avatar
11 votes
0 answers
786 views

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 $C^{\circ r}(n)$ ...
Sebastien Palcoux's user avatar
15 votes
1 answer
1k views

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 ...
Yuzuriha Inori's user avatar
5 votes
2 answers
1k views

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, $...
ReverseFlowControl's user avatar
11 votes
1 answer
2k views

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 ...
Wojowu's user avatar
  • 27.7k
2 votes
1 answer
423 views

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 = \{...
vassilis papanicolaou's user avatar
3 votes
1 answer
2k views

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 ...
Dominic van der Zypen's user avatar
8 votes
1 answer
256 views

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 ...
Noah Giansiracusa's user avatar
33 votes
2 answers
6k views

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 ...
Tanner Swett's user avatar
  • 1,153
-1 votes
2 answers
369 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&...
MathAllTheTime's user avatar
8 votes
3 answers
455 views

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,...
Igor Pak's user avatar
  • 16.8k
6 votes
1 answer
482 views

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 ...
Dan Brumleve's user avatar
  • 2,264
4 votes
0 answers
356 views

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 ${\rm C}^{\star}(X)$, ...
Sebastien Palcoux's user avatar
9 votes
1 answer
2k views

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 ...
Sebastien Palcoux's user avatar
3 votes
2 answers
428 views

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 ...
JMJ's user avatar
  • 263
12 votes
3 answers
2k views

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 ...
Ofra's user avatar
  • 1,613