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
3
votes
1answer
596 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 ...
1
vote
0answers
305 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 $(...
2
votes
0answers
194 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 ...
10
votes
1answer
418 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 ...
3
votes
0answers
319 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 ...
6
votes
1answer
614 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 ...
2
votes
1answer
231 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 ...
0
votes
0answers
272 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 ...
5
votes
1answer
778 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 ...
5
votes
0answers
187 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: $$ T(n) = \left\{ \begin{array}{} 3n+1, & \text{if $n$ is odd}\ \\ n/2, & \text{if $n$ is even} \end{array} \right. $$ ...
4
votes
2answers
283 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 ...
2
votes
0answers
317 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 $n$ ...
25
votes
3answers
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 & \...
7
votes
0answers
473 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 $C^{\circ r}(n)$ ...
15
votes
1answer
690 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 ...
5
votes
2answers
902 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, $...
13
votes
1answer
991 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 ...
2
votes
1answer
348 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 = \{...
3
votes
1answer
1k 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 ...
8
votes
1answer
219 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 ...
21
votes
2answers
3k 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 ...
-1
votes
2answers
346 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&...
8
votes
3answers
336 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,...
4
votes
0answers
227 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 ...
7
votes
1answer
958 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 ...
3
votes
2answers
333 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 ...
11
votes
3answers
1k 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 ...
0
votes
1answer
263 views

Are there infinitely many solutions of $2^k=3^z-1$ with $k,z \in \mathbb{N}$? [duplicate]

This question arose as an attempt to answer the following question Relaxed Collatz 3x+1 conjecture. I wanted to show that there is a solution of the equation $2^{k}=3^{z}(2n+1)-1$ for each $n\geq 2$, ...
0
votes
1answer
339 views

A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?

Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
30
votes
2answers
1k views

A Collatz-like problem on prime numbers

Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
22
votes
2answers
763 views

Possible behaviors of integer sequences that arise from powering nonnegative integer matrices

Let's call a sequence of nonnegative integers $x_1,x_2,\ldots$ matrix-realizable, if there exists a $k\times k$ nonnegative integer matrix $A$ (for some $k$), as well as nonnegative integer vectors $u,...
13
votes
3answers
1k views

Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...
17
votes
2answers
928 views

Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (in June 2016): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \...
12
votes
1answer
1k views

Some Questions on the Collatz conjecture (reexpressed as “equivalence relation”)

The set of all positive whole numbers is denoted by $\mathbb{N}_+$. Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto \begin{cases}\frac{n}{2}&\text{$n$ even}\\3n+1&\text{$n$ odd}\end{cases}$...
43
votes
1answer
3k views

Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can eventually be reduced to $1$ by iterative application of the maps $x \mapsto 3x+1$ whenever $x$ is odd and $x \mapsto x/2$ whenever $...
14
votes
1answer
369 views

A Collatz-like question about permutations

An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting. Consider all permutations $\pi$ on the natural numbers such that ...
8
votes
1answer
738 views

Residue class sufficiency sets for the Collatz conjecture

I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of ...
44
votes
1answer
2k views

Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
6
votes
0answers
400 views

Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes: $f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is prime} \\ \lfloor n/2 \rfloor & \text{if} \;n \;\text{...
22
votes
3answers
2k views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
-1
votes
1answer
352 views

Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.) a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value. consider a ...
5
votes
2answers
1k views

3n+1 problem and cycles

Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). ...
3
votes
0answers
163 views

Largest permutation groups without “non-mixing” subgroups

We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 ...
8
votes
1answer
3k views

Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
1
vote
1answer
220 views

Group with 2 orbits on the nonnegative integers — description of the orbits

Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}...
19
votes
3answers
1k views

A curious sequence of rationals: finite or infinite?

Consider the following function repeatedly applied to a rational $r = a/b$ in lowest terms: $f(a/b) = (a b) / (a + b - 1)$. So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$. I am wondering if it is ...
9
votes
1answer
1k views

Collatz stopping-time and Poisson distribution, and connection to other problems?

I read many threads about Collatz here - so don't worry, this is no attempt to any proof, just asking about a curious fact: This graph gives the stopping-time of Collatz sequences up to $n=10^8$ (...
-3
votes
1answer
657 views

Point me to an attempt to Proove Collatz Conjecture by Substitution and Factor analysis? [closed]

Summary of Question: Where can I find a discussion of attempting to prove the Collatz Conjecture via substitution and abstract examination? I've done a lot of reading on the problem, including ...
0
votes
1answer
1k views

Implication for cycles (of some length $m$) in Collatz-type problems: typical ratio between largest and smallest element?

Background Consider Collatz-type problems of the form $an + 1$, where $a > 2$ is a positive, odd integer (e.g., $3n + 1$, $5n +1 $, $7n + 1$, etc.). For convenience, automatically divide by two. ...
9
votes
3answers
3k views

Larger cycle than 4, 2, 1 in Collatz iteration?

(Here I discuss the Collatz problem only for positive integers.) It is possible, by computation, to find all cycles in the Collatz iteration of a fixed length. It is clear that an increase must be ...