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.
353
questions
2
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 = \...
0
votes
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 ...
0
votes
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$...
-2
votes
2answers
85 views
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 ...
0
votes
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=...
-1
votes
0answers
86 views
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 ...
0
votes
0answers
59 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, ...
0
votes
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?
4
votes
0answers
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$$...
2
votes
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 ...
0
votes
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 ...
1
vote
0answers
89 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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
4
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
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}$$
where k, l, m, n, p can only be natural numbers >=1.
The unknown t has also to be a natural number >= 1.
...
1
vote
1answer
90 views
What's wrong with this reasoning concerning collatz cycles?
Say there exists a collatz cycle: $k_0, k_1...k_{m-1}, k_0$
This means that:
$$3^a k_n + \sum^{a-1}_{i=0} 3^i*2^{b_i} = 2^c k_n $$
where $b_i > b_{i+1}$ and $b_{a-1} = 0$ and since there are no ...
0
votes
1answer
60 views
Collatz Cycles with double the even numbers than odd
This question is with reference to the collatz conjecture. It is known that a number $A$ in a collatz cycle is of the form:
$$A = \sum_{i=1}^k \frac{ 3^{k-i} \cdot 2^{k_i}}{2^n - 3^k},$$
$0 = k_1< ...
5
votes
1answer
930 views
Calculate the maximum in the Collatz sequence
Consider the notorious Collatz function
$$ 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 ...
7
votes
0answers
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 ...
-1
votes
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:...
2
votes
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 ...
-1
votes
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 ...
4
votes
0answers
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 ...
-1
votes
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 ...
1
vote
0answers
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,
$...
2
votes
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 : \...
0
votes
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}$$
Define $f^x$ as the $x$'th iterate of $f$....
0
votes
1answer
55 views
Natural functions that converge to 1
What are some examples of functions that given a positive integer x converge to 1 using only natural operations (add, sub, div, mul).
An example would be the collatz conjecture, although it requires ...
2
votes
1answer
154 views
Ratio of (Collatz “path” bits) / (starting number bits) $\approx 2$
The "Path Records" page by Eric Roosendaal related to the Collatz conjecture (see here), reports a "striking" feature of the path records.
If we define $\text{MaxSeq}(M)$ (in Roosendaal it is $\text{...
-1
votes
1answer
62 views
How do I correctly (or better) define this bound or constraint in the padic ring $\Bbb Z_2$?
Let $T(x)$ be a partial function $:\mathcal P(\Bbb Z^+)\to\Bbb Z_2$.
I have the rule for a certain $\underline x\in \mathcal P(\Bbb Z^+)$ that at least every fourth digit of $T(\underline x)$ is a $1$...
1
vote
2answers
168 views
Do I understand the concept of $x^{0.84}$ correctly?
I'm trying to understand the concept of $x^{0.84}$ that Jeffrey Lagarias found for Collatz Conjecture. If I'm wrong, please correct me with an answer. I understand such that,
Suppose the interval $\...
2
votes
0answers
94 views
Distributed computing project to check the convergence of the Collatz problem
I am about to start a distributed computing project to check the convergence of the Collatz problem.
So far, the convergence of the problem has been verified for all numbers below $87 \times 2^{60} \...
1
vote
1answer
157 views
Is there a Collatz 'branch' stopping time equation?
Definitions:
-Collatz: odd: $\frac{3x+1}{2}$ | even: $\frac{x}{2}$
-Branch: Starting at an odd number and increasing until reaching an even number. Then decreasing until reaching another odd number (...
2
votes
3answers
152 views
The $k^{th}$ iteration of $2^k-1$,$2^k-3$ ,and $2^{k}-11$ and others
With the short cut version of the $3x+1$ conjecture, $$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^{...
2
votes
0answers
92 views
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 $$1\to 2 \to 1\to 2.....
2
votes
0answers
314 views
Showing $3^iq-2^{i-p}\neq2^pq-1$, with $p:=\lceil i\,\log_23\rceil$, $q:=\left\{\frac{2^{i}\,3^{2^{p-i-2}}}{2^p3^i}\right\}$, $i>6$
For this inequality
$$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\...
3
votes
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,
...
-1
votes
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 ...
5
votes
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 "...
0
votes
0answers
69 views
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 ...
2
votes
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 ...
7
votes
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 ...
0
votes
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 ...
1
vote
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;...
5
votes
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
votes
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 ...
