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.

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Question re: Collatz Conjecture

Disclaimer: I'm not a mathematician and for the most part I do not understand what, exactly, it takes to mathematically prove anything. That is where this question comes from. Why can't the Collatz ...
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Is this single Collatz rule useful?

I have thought of a single simple rule for Colatz conjecture that is slightly different to the normal one. Here it is: $$f:(a,b)\rightarrow (3a+\gcd(a,b),b)$$ Then the Colatz conjecture can be written ...
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Collatz conjecture : Is that the reason why 3x+1 forms loop with some seeds?

Lets start with a negative number: -7 $$ -7\times3+1=-20\\ -20\div2=-10\\ -10\div2=-5\\ -5\times3+1=-14\\ -14\div2=-7 $$ we can sum up these steps as the following functions: $$ -7\times3+1=-20\\ {-7\...
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An approach to solving $2(2^{(a+b+c+2)} + 3\cdot 2^{(a+b+1)} + 9\cdot 2^a + 101)$ for multiples of 77

In searching for counter examples to the Collatz Conjecture, I have come across a family of equations with the following form. $$77D = 2(2^{a+b+c+2} + 3 * 2^{a+b+1} + 9 * 2^a + 101)$$ Where $$0>a+...
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61 views

Collatz, better terminology than "total stopping time"?

In the Collatz conjecture, the total stopping time is the number of iterative applications of the collatz map on a number n that is needed to get to the result 1. Now, the total stopping time is not ...
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2 votes
1 answer
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"3n + 1 problem" variant, using (n/2 + 3)

Background Variant "3n + 1 problem" Consider a mapping similar to the original "3n + 1 problem" (Collatz conjecture): $n → n/2 + 3$ for even n (variant rule due to +3) $n → 3n + 1$...
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1 vote
1 answer
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Can someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?

Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '...
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1 answer
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Is it possible to find closed form of the Collatz conjecture? [closed]

So recently I was thinking about converting the recurrent definition of Collatz conjecture into a closed-form expression, which would map any $n,\space n\in\mathbb{N}$ to the $n^{th}$ iteration of the ...
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19 votes
2 answers
662 views

Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?

Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ...
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6 votes
3 answers
360 views

Why does this cycle of 44 show up in the Collatz Conjecture?

Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{...
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  • 141
16 votes
1 answer
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Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle 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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1 vote
1 answer
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Is there a name for this function or a concept similar to it?

I'm wondering if anyone has heard or seen a function that looks or behaves like this one. $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{...
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  • 141
0 votes
1 answer
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A typo in an implementation of the Collatz algorithm made it loop infinitely

This question is kind of related to programming but my question only relates to math in a theoretical sense. I was trying to implement a Collatz conjecture algorithm that calculates the steps it needs ...
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2 answers
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Is there an explanation for clustering of total stopping times in Collatz sequences?

I assume knowledge of the Collatz conjecture. Here I'm looking only at total stopping times $t(n)$, and mostly will drop the word "total." Just looking at relatively simple graphs of ...
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Collatz Conjecture and non-trivial cycles

Consider $$n \rightarrow ... \rightarrow an+b$$ to be a sequence of natural numbers to which we apply $3n+1$ or $\frac{n}{2}$ operations. This sequence is called a cycle, if and only if $an+b=2n$. (...
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1 answer
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Has anyone discovered this Collatz Conjecture pattern?

I noticed that there is a linear increase of +1 in the stopping times of a sequence of numbers with the seeds being the sum of the previous number in the sequence added to itself. I tried this on ...
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3 votes
2 answers
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For any composition sequence $s$ of maps $h(X)=X/2, \ f(X)=(3X + 1)/2$, there exists an integer $X$ such that its Collatz sequence contains $s$

Let $h(X) = X/2$, and $f(X) = (3X + 1)/2$. Then clearly every iteration $g^i(X), X \in \Bbb{Z}$ the Collatz mapping $$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \ X = 1\pmod 2 \end{...
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1 answer
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Terras lemma proof

For any number N the parity vector v(N) is defined as $v_i(N) = S_i \pmod 2$ If N is a positive integer of the form $a\cdot 2^k + b (b < 2^k)$ then the first k elements of the parity vector are ...
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1 vote
1 answer
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My proof that there exist no odd 2-cycles for the accelerated Collatz function other than $(1,1)$. Do you have a link to a historical proof?

Let $f(x) = \dfrac{3x + 1}{2^{\nu_2(3x + 1)}}$. Clearly the $(1,2,4)$ cycle from the original Collatz map disappears under this context. So the Collatz conjecture-equivalent goal with $f$ is that ...
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2 answers
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If there are $k_1, k_2, \dots, k_n$ divisions by $2$ in a Collatz cycle, then $k_1 + \dots + k_n \geq n$, but can we get a greater lower bound?

Let $f(x) = |3x + 1|_2(3x + 1)$ be the accelerated Collatz function, where $|3x + 1| = 2^{-\nu_2(3x + 1)}$ is the $2$-adic absolute value. Clearly for all $x$ odd we have $\nu_2(3x + 1) \geq 1$ so ...
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7 votes
1 answer
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The Collatz Conjecture function should induce a collection of Grothendieck groups, one for each $n \in \Bbb{Z}$ or $\Bbb{N}$. Their properties?

This question is about the Collatz conjecture. Let $\Bbb{N}$ include $0$. The Collatz conjecture function is given by: $$ f: \Bbb{N} \to \Bbb{N}, \\ f(n) = \begin{cases} \dfrac{n}{2}, \text{ if } n = ...
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3 votes
1 answer
426 views

Status of the Nichols claim to proof of the Collatz conjecture? [closed]

In December 2021, Robert Hills Nichols, Jr of Cumberland University claimed a proof of Collatz on arXiv. But I've not seen any reviews or comments on the validity of his claim, and it's certainly ...
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How to know when the Collatz conjecture has been proved?

When using google to find out about research results about the Collatz conjecture, I find numerous proofs by various people who seem to be experts of the topic and an abundance of proofs by amateurs. ...
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2 votes
0 answers
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What progress has been made on the Collatz conjecture since Crandall's 1978 paper?

I have recently read the famous paper by Crandall (1978) on the 3x+1 problem, and I wonder what progress has been made since then. The paper claims that: If a cycle exists, then the minimum number of ...
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3 votes
3 answers
715 views

Collatz proven for primes = proven for all integers?

I recently saw a video stating that if the Collatz conjecture was proven for prime numbers, it was proven for all numbers. That's the first time I see this and I can't find any references with the ...
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0 votes
1 answer
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A Collatz generalization and approximation of a bounded-unbounded point of Collatz-like Functions

I have been working on Collatz-like functions to test the probabilistic heuristic argument in favor of all trajectories being bounded (ending in a cycle) and have come up with a generalization to test ...
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Can there be integer solutions (please PROVE) [closed]

With the following equation- $\frac{(3^x-2^x)}{(2^{(x+y)}-3^x)} = z $ What $x,y,z$ (all integers) satisfy this? The trivial solutions are- $X=1,y=1,z=1 ,X=2,y=1,z=-5 ,X=1,y=0,z=-1 $ This is my ...
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18 votes
3 answers
902 views

Is it already known that $\sum_{i=1}^x\cos(S(i))\sim ax\cos(b\ln x)$, as $x\to\infty$, where $S(i)$ is the number of Collatz steps from $i$ to $1$?

I was playing with the Collatz Conjecture today, and empirically found a curious behaviour: Let $S(i)$ be the function that calculates the number of steps needed for $i$ to reach $1$: It seems that $\...
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2 answers
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What exactly is a Collatz-like Problem

What exactly is a Collatz-like problem? Let $f:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1: $$f(n) = \begin{cases} n/2, & \...
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4 votes
1 answer
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Representing Collatz sequences with Pascal's simplices modulo 2?

In this binomial coefficient pdf (from a Russian summer school?), I found a curious identity (page 2, solution on page 10) pointed out by the authors, which ties the modulo 2 Pascal's triangle (first ...
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4 votes
2 answers
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Veritasium Collatz Video - Clarification About Loops

I've watched the recent video about Collatz by Veritasium. At one point in the video (11:17) he mentions that if you can show that for every seed value, there is at some point a number less than the ...
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1 vote
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Additional assumptions for reducing the complexity of the collatz sequence.

in addition to the original transformation rules stated in the collatz conjecture ,in the passage given below the rules are tweaked a bit and some additional rules are added leading to the generation ...
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2 votes
2 answers
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what is so special about the number $3$ in the $3n+1$ conjecture?

Recently, I was quite intrigued by the $3n+1$ conjecture and it left me wondering what is so special about the number $3$? With the same rules apply, does there exist another positive integer other ...
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4 votes
0 answers
265 views

Will repeatedly applying the following operation always leads to an odd number divisible by $3$

Let: $x_i > 1$ be any odd integer that is not divisible by $3$ $\nu_2(x)$ be the 2-adic valuation of $x$ $x_{i-1} = \begin{cases} \text{undefined}, && \text{if }x_i\equiv 0 \pmod 3\\ \...
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3 votes
1 answer
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Tennenbaum's theorem generalizations: can we have a computable Collatz map on a nonstandard model of Peano?

Tennenbaum's theorem states that for any (countable) nonstandard model of Peano arithmetic, neither the addition nor the multiplications is computable. I find this result fascinating (and frustrating!)...
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2 votes
2 answers
205 views

What are the known rules on the seed that guarantee the Collatz sequence will stop

I have written some code that searches for counter-examples to the Collatz conjecture. The following fact enables me to accelerate the search. In the paper "Garner, Lynn - On The Collatz 3n + 1 ...
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4 votes
0 answers
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Collatz Related? Are there any generalized rules for the following?

Following up on my last question, Last Question. I have also noticed the following: If $x \bmod 5=0$ : execute $x/5$, or elseIf $x$ ends in $1$ : execute $(x⋅2)+3$, or elseif $x$ ends in $3$ : execute ...
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2 votes
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For calculating Collatz class records, how is the sieve constructed and used?

I'm looking at the class record algorithm here. I understand how some of the numbers (e.g. 8*k+5) can be skipped because they join with a lower number. But I don't understand exactly how the sieve of ...
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4 votes
1 answer
166 views

Collatz Related? $x$ is odd: If $x \bmod 3=0$ execute $x/3$, or elseIf $x \bmod 3=1$ execute $(x⋅2)+1$, or elseif $x \bmod 3=2$ execute $(x⋅4)+1$.

Let $x$ be an odd number: If $x \bmod 3=0$ : execute $x/3$, or elseIf $x \bmod 3=1$ : execute $(x⋅2)+1$, or elseif $x \bmod 3=2$ : execute $(x⋅4)+1$. I have been playing around with my calculator for ...
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4 votes
3 answers
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Does the following Collatz-like bijection have divergent trajectories?

Let $f:\mathbb Z\rightarrow\mathbb Z$ be the function defined by $$f(3n)=2n$$ $$f(3n+1)=4n+1$$ $$f(3n+2)=4n+3$$ Is there any $x\in\mathbb Z$ such that the sequence $x,\,f(x),\,f(f(x)),\,f(f(f(x)),\...
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4 votes
0 answers
260 views

Random search for very big Collatz conjecture counter-examples

I know that exhaustive search was done to test numbers up to 2^68. This seems like a big number but when looking at Collatz function as a Turing machine manipulating some input bit sequence, only ...
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19 votes
0 answers
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Smallest $m>1$ such that the number of Collatz steps needed for $238!+m$ to reach $1$ differs from that for $238!+1$.

Let $h(x)$ be the number of steps^ needed for $x$ to reach $1$ in the Collatz/3n+1 problem. I found that $$h(238!+n)=h(238!+1), \;\; \forall 1 < n \leq 690,000,000$$ Here "!" is the ...
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4 votes
0 answers
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What is known about the base 3 string version of Collatz conjecture

I can equate Collatz conjecture's function to: If a string has an odd number of 1's append a 1 If a string has an even number of 1's : a) copy the position of the 2's into a new 0 only string, and ...
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0 votes
2 answers
238 views

Why does this “proof” for the Collatz conjecture not work? [closed]

After seeing a video on the Collatz conjecture, I kind of played around with it a bit and found the following result (which can be shown by induction): $$\underbrace{f \circ \ldots \circ f}_{N \text{ ...
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6 votes
2 answers
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What does it take to find integer solutions to this exponential division equation?

Consider this equation where $a$ and $b$ are positive integers. $$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$ This equation has the trivial solution $k=1, a=1, b=1$. How would I find more solutions, or show ...
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0 votes
0 answers
224 views

Jacobsthal numbers occurring in reduced cobweb plot for the Collatz Problem

I have been working on the Collatz problem for a while now, and have made this efficient cobweb plot function for it, where it automatically does all the dividing by two and always returns an odd ...
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1 vote
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Collatz conjecture but with $\ 3n-1\ $ instead of $\ 3n+1.\ $ Do any sequences go off to $\ +\infty\ $?

Collatz conjecture but with $\ 3n-1\ $ instead of $\ 3n+1.\ $ Do any sequences go off to $\ +\infty\ $? $$$$ Background (not necessary to answer my question): Considering the following operation on ...
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4 votes
2 answers
392 views

Collatz Conjecture: Reasoning about the possible divergence of a collatz sequence

I've been thinking about the possibility of a divergent collatz sequence for the Collatz Conjecture. In other words, the possibility that neither a trivial nor non-trivial cycle is ever reached. If ...
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3 votes
0 answers
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What does Terry Tao mean by exponential separation

Tao wrote in his blog post (in the fourth last paragraph): Thus we see that any proposed proof of the Collatz conjecture must either use transcendence theory, or introduce new techniques that are ...
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9 votes
2 answers
438 views

Collatz Conjecture: For a cycle where the maximum odd integer is $x_{max}$, does it follow that $x_{max} < 3^n$

I am working on understanding the upper limit in the case where a non-trivial cycle exists for the Collatz Conjecture. Is the following reasoning valid for establishing that the maximum odd integer in ...
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