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.

Filter by
Sorted by
Tagged with
0
votes
2answers
163 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
438 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 ...
3
votes
3answers
244 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
825 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
138 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
87 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
908 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
358 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 ...
0
votes
2answers
232 views

Solving Collatz Conjecture through analysis of their Binary Representations [closed]

Collatz conjecture is generalized by the statement: "For any number greater than zero, if the number is even then divide it by 2. If the number is odd, multiply the number by 3 then add 1." The ...
3
votes
1answer
108 views

Questions on a formula for the Mertens function

The Mertens function $M(x)$ is defined as follows. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)$ I've noticed the Merten's function can also be evaluated as follows which is related to OEIS entry ...
1
vote
1answer
168 views

Is $f$ a continuous function?

I seek to understand whether a certain discontinuous function on the dyadic rationals can be recast using the Cantor set as a continuous function. Let $X$ be the dyadic rationals in the interval $(\...
4
votes
3answers
504 views

Proof of Bound for Growth of Divergent Trajectory in $3x+1$ Problem

In this paper, Lagarias makes the following claim in section 2.7 (Do divergent trajectories exist?). Context $$T(x) = \left\{ \begin{array}{rl} \dfrac{3x + 1}{2}, & 2 \nmid x \\ \dfrac{x}{2}, &...
0
votes
0answers
60 views

Switches (binary) and dominoes in a “Collatz-universe”

Imagine in a "Collatz-universe" that is discrete and a complex system is buildt up of switches that can either be on or off. How can I make a comparison between physical switches and dominoes in ...
1
vote
0answers
251 views

Collatz conjecture, Tao-Collatz remainder and mod n.

Collatz conjecture is equivalent to $n\times 3^{k} = 2^{ak+1} - TCR$ where, for me, $k$=odd steps, and $ak+1 $=even steps. Note that total steps = k +( ak+1) steps. Some numbers have the same total ...
-3
votes
1answer
92 views

Polynomial extension of Collatz graph… does it converge finitely?

Let $\Bbb Z[2^{-1},2^{-1/2},2^{-1/4,\ldots}]$ be the ring of dyadic rationals extended to include dyadic powers of $2$. Then let $2^{\nu_2(x)}$ extend the 2-adic valuation to dyadic powers of $2$ (...
-1
votes
1answer
132 views

Found a formula for generating all numbers in all possibilities of all cycle lengths

yo, I'm about to spread some new knowledge about the collatz conjecture. Not sure if this has been shown before or not, but here: https://en.wikipedia.org/wiki/Collatz_conjecture#Cycles it states that ...
0
votes
3answers
153 views

a question about the Collatz conjecture (relation of smallest number in a cycle and minimal cycle-length)

I've only done a bit of research on the current findings, not sure if anyone here can answer this. Q1: I just haven't been able to find, has it been shown yet that a it is ...
1
vote
1answer
50 views

Infinite Number of Infinite Fractions From Hailstone Sequences? (Non-Optimized Collatz)

Starting with some number, you can generate the hailstone sequence from it. In the case of 3, the (finite) hailstone sequence of it is $[3,10,5,16,8,4,2,1]$. Placing them in a continued fraction like ...
5
votes
3answers
665 views

Where is my Collatz conjecture proof wrong?

I am amateur and don't have very good understanding of mathematical proving. My proof is so simple i don't believe noone thought of this before. But I am so blinded by the hope it is correct, that I ...
0
votes
0answers
40 views

Are there nontrivial subsets of the complex unit circle satisfying the multiplicative Jacobi identity?

Let $a^{(b^c)}\times b^{(c^a)}\times c^{(a^b)}=1$ Then a set $S$ satisfies the multiplicative Jacobi identity if this is true for all $a,b,c\in S$. $S=\{1\}$ satisfies the identity. $S=\{0\}$ doesn'...
0
votes
1answer
90 views

How do p-adic fields degenerate for non-prime $p$?

How do p-adic fields degenerate for non-prime $p$? Let $d(x,y)$ be the inverse of the highest power of $4$ that divides $\lvert x-y\rvert$ Then let $\Bbb Z_4$ be the completion of $\Bbb Z$ under ...
4
votes
3answers
132 views

Collatz conjecture undecidable from the general case?

There are known results that generalized version of Collatz conjecture is undecidable. I wonder why special case of it still can be decidable? Isn't general case should apply results to all special ...
0
votes
1answer
59 views

Bitlength probabilities of the shortcut Collatz map

If my calculations are right the probability that the next number in Collatz'sequence has increased its bitlength is equal to $31.25 \%$ Bitlength meaning; the length from a fixed position of least ...
1
vote
2answers
126 views

Collatz conjecture: $2^{m-1}(6n-3)$ is not part of any cycle

My original method was different from the method shown here. Instead of working my way backward through the iterations as below, I worked my way forward. I choose against doing that here despite of it ...
-2
votes
1answer
92 views

Help with work on collatz conjecture?

I've managed to rearrange the collatz conjecture into a formulae and was wondering if I'm going in the right direction? The Formulae: $y=\left(3\left(\operatorname{mod}\left(x,2\right)x\right)\right)...
1
vote
2answers
97 views

Why does plotting $\frac{3n+1}{2^k}$ give several apparently straight lines?

Here's the plot for $f(n)=\frac{3n+1}{2^k}$ ($n$ integer, positive, odd) where $k$ is the number of 2 factors in $3n+1$: Just to be clear, here's a zoom in: Why do we see those straight lines? I ...
2
votes
1answer
296 views

New record for for lower bound of non-trivial cycle lengths of Collatz sequences?

This is an indirect follow-up of the previous post I did on the Collatz conjecture. After a few responses, we managed to get to the fact that if we have $n\in\mathbb N$ and cyclic $(e_n)$ such that $...
1
vote
1answer
166 views

What is the sequence of accumulation points in the 2-adic space, of the Collatz graph?

In the orbit of the function $3x+2^{\nu_2(x)}$ through "accumulation points" of the Collatz graph I have: $?\mapsto\dfrac{-\langle2\rangle\cdot\{5,7\}}{9}\mapsto\dfrac{-\langle2\rangle}{3}\mapsto \...
1
vote
2answers
253 views

What formula could generate this sequence related to the Collatz conjecture

The collatz conjecture states that every number eventually reaches $1$ under the repeated iteration of $$ f_0(n) = \begin{cases} n/2, & \text{if $n$ even} \\ 3n+1, & \text{else} \end{cases}$$...
16
votes
3answers
723 views

A possible way to prove non-cyclicity of eventual counterexamples of the Collatz conjecture?

I've been recreatively working on the Collatz conjecture for a few months now, and I think I may have found something that could potentially prove at least half of the conjecture, which is the non-...
3
votes
1answer
78 views

Cycles in bit-limited verson of Collatz map

By 'bit-limited', I mean that we have a computer/calculator that handles binary numbers up to $n$ bits ($n>=1$), and any numbers greater than $2^n-1$ overflow by truncating the higher bits. ...
2
votes
3answers
142 views

Number of Collatz steps for Mersenne numbers

I noticed that for all $k \in \mathbb{N} \geq 1$ the following is true (I tested up to $2^{5000}$): $\text{Collatz_Steps}(2^{2k+1} - 1) + 1 = \text{Collatz_Steps}(2^{2k+2} - 1)$ Where $\text{...
1
vote
2answers
194 views

Irritating “proof” of the Collatz Conjecture

I recently stumbled across this self-proclaimed proof of the Collatz Conjecture. It seems very irritating to me that this very hard conjecture is supposedly proven by using very basic counting ...
0
votes
1answer
80 views

$3n+1$ graph and shared vertices

I found this somewhere on the web: Theorem. The total number of vertices for $n$ squares that share exactly one common vertex is given by the formula $f(n) = 3n + 1$. Proof. Each of the $n$ squares ...
-3
votes
2answers
488 views

Preserving historical information of the Collatz function?

In some sense this two equations are the same, namely $f_2$ preserves the historical information of $f_1^n$, where the exponent is function composition, but I am not sure how to show this rigorously. $...
1
vote
1answer
160 views

How to prove property of Collatz Conjecture trees (3x+1 Problem)

Inevitably for any amateur mathematician, I've been playing with the Collatz Conjecture. I have found it's easier to examine and to generate theorems if we use this equivalent statement: $$c_{n+1}\...
0
votes
1answer
148 views

"What is the specific mathematical reason behind the origin of the Collatz Conjecture that makes it difficult to solve it? [closed]

Is there a known so specific mathematical reason that makes it difficult to solve the Collatz Conjecture? Clearer: What is the specific mathematical reason behind the origin of the Collatz ...
0
votes
1answer
452 views

Why is the Collatz Conjecture so difficult to prove or disprove? [duplicate]

Is there some quality of the Collatz Conjecture that has made it so difficult to prove or disprove? Besides just using a computer to calculate lots and lots of values of $n$, of course.
1
vote
2answers
118 views

Collatz $2x + 1$ conjecture?

Do we know of any Collatz theorem involving similar functions. For example what do we know about iterations of: $$ f(x) = \begin{cases} \dfrac{x + 1}{2} \text{, if } x \text{ is odd}. \\ 2x + 1, \...
1
vote
1answer
150 views

Polynomial divided by a monomial in some extended Collatz Shortcut function

Quirks of the Collatz shortcut function. One example and some basic questions. The functions here are defined as: $$C_d(n):\mathbb{Z^+}\rightarrow\mathbb{Z^+}$$ $$n\in\mathbb{Z^+}$$ The definition ...
3
votes
2answers
172 views

Is there a polynomial (or series) expression for summing $S_d(a,N)=\sum_{k=0}^{N-1} \log(1+{1\over a+k \cdot d})$? (perhaps Bernoulli-type)

I need a quickly evaluatable expression for sums of consecutive logarithms of the type $$ S_{d}(a,N) = \log(1+ {1\over a})+\log(1+ {1\over a+d})+\log(1+ {1\over a+2d})+ \cdots + \log(1+ {1\over a+(N-1)...
4
votes
3answers
214 views

$k$-cycles in Collatz functions

I have a couple of questions, but I need to give some quote and some reasoning before I ask. Quote from Wikipedia: A $k$-cycle is a cycle that can be partitioned into $2k$ contiguous ...
3
votes
1answer
137 views

There are lower bounds worked out for the length of nontrivial Collatz-cycles. How can *upper bounds for the disproof* be determined?

There have been lower bounds estimated for the length $N$ of (odd) steps of a nontrivial cycle in the collatz-problem. Such estimates have been based on knowledge of upper bounds $\chi$ for any number ...
3
votes
2answers
556 views

What is currently the highest lower bound for the length of a nontrivial cycle in the Collatz Conjecture?

We know that there are two possibilities to disprove the Collatz Conjecture. We find a nontrivial cycle. We find a sequence that diverges to $\infty$ A non-constructive disproof is imaginable as ...
1
vote
2answers
202 views

Literature request: Papers by Charles C. Cadogan.

I am looking for three papers on the Collatz Conjecture by Charles C. Cadogan. The Annotated Bibliography II (Lagarias) lists these three as: "Charles C. Cadogan (2000), The 3x+ 1 problem: towards a ...
2
votes
2answers
132 views

Does anyone know where I can get a copy of R. P. Steiner, “A theorem on the Syracuse problem”?

R.P. Steiner. "A theorem on the Syracuse problem". In: ed. by D. McCarthy and H. C. Williams. Congressus numerantium; 20. Proceedings of the 7th Manitoba Conference on Numerical Mathematics and ...
2
votes
1answer
100 views

Is this Algorithm's Result concerning Collatz Sequences provable?

I have searched the www for this little algorithm and its result thoroughly, but it was nowhere to be found, do you know if my observation is provable? The Algorithm: Step 1: Select an arbitrary ...
1
vote
3answers
132 views

Is there a maximum number of consecutive decreasing steps a Collatz cycle can have?

If we take a look into the (known) cycles of the Collatz Conjecture when all integers are included, we get 4 cycles: 1 → 4 → 2 → 1 … −1 → −2 → −1 … −5 → −14 → −7 → −20 → −10 → −5 … −17 → −50 → −25 ...
0
votes
0answers
49 views

What terminology should I use when refrencing how close a sequence is to a loop for research?

I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping. For example, If I was interested in this ...

1
2
3 4 5
8