Questions tagged [collatz-conjecture]
For questions about the iterated map $n \mapsto 3n+1$ if $n$ is odd and $n \mapsto \frac n2 $ if $n$ is even, and its generalizations.
561 questions
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Is there a formula to generate all n - step odd numbers that all reduce to 1?
I had a quick question about the Collatz Conjecture. According to the function in which the Collatz Conjecture is defined, all even numbers must necessarily turn into an odd number. Therefore, it ...
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Convergence of infinite series generated by inverse Syracuse functions
NOTE: While this question was inspired by me playing with the Collatz Conjecture, it is not related to the Collatz Conjecture directly. If I should remove the tag, please let me know.
Define: $N = \{n ...
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Existence of infinite sequence satisfying Collatz-style function [closed]
Consider the following sequence:
$x_0 = x$ where $x$ is a positive integer
For all $n \geq 1$, $x_n = f(x_{n - 1})$, where
$f(k) =$
\begin{cases}
\frac{a_0k + a_1}{c} & \text{if } k \equiv \...
8
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1
answer
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Avoid unnecessary calculations when multiplying matrices if only need one element of resulting matrix
The Problem:
I need only the bottom left element of a product of matrices $(\bf{M_1}+\bf{I})(\bf{M_2}+\bf{I})\cdots(\bf{M_N}+\bf{I})$,
where $\bf{I}=\begin{pmatrix}
1 & 0\\
0 & 1\end{pmatrix}$,...
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Status of Collatz Conjecture for Negative Integers (a.k.a. $3x-1$ problem)
I was (surprisingly) not able to find online much information about the Collatz Conjecture on the negative integers. We know there are three cycles: $(-1 \rightarrow -2 \rightarrow -1)$, $(-5 \...
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answer
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Roots of polynomials defined by the Syracuse (Collatz) sequence
I'm a french graduate student, and I stumbled on a problem which seems to surpass my current abilities...
My goal was to study polynomials defined by the Syracuse sequence (or Collatz sequence). By ...
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Redefining the Meta-Turing Problem by Introducing an pinfHalt State
Abstract:
This proposal aims to approach the Turing Halting Problem by introducing the concepts of non-computable functions and an additional state, "pinfHalt," which denotes a program that ...
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answer
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Does the shape of the collatz tree exhibit self-similar properties?
Note: Throughout this question I use the term "branch" to mean a part of the Collatz tree beginning with an odd number $n$ and then consisting of $n \cdot 2^k$ for all $k$.
My question ...
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If the Collatz conjecture is undecidable, then it is true
Suppose that the Collatz conjecture is undecidable in PA. Then, by Godel's completeness theorem, there are models where it is true, and models where it is false. Let M be a model where it is true. The ...
user1381060
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Is there a way to figure out the behavior of hailstone sequence with binary representation?
I'm a university student and posting this question while studying $3n+1$ conjecture.
Recurrence relation of sequence is as follows:
$$ a_{n+1} = \begin{Bmatrix} \frac{a_n}{2},& \text{if }a_n \...
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Collatz sequence but multiplying by a large odd number rather than $3.$ What is the simplest way to prove that a sequence goes off to infinity?
Under the Collatz rules:
$n\to 837n+1$ if $n$ is odd
$n\to n/2$ if $n$ is even.
What is the simplest argument/proof to show that there is a Collatz sequence with a starting number that goes off to ...
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7 for 5n+1 series diverges to infinity?
Write an odd integer in the "modified" binary form as $2^m - 1$ and apply 5n+1 (similar to applying the rules of original 3n + 1) to it:
\begin{align}
n &= 2^m - 1 \\
...
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Reasoning about the Collatz conjecture, multiple infinitely growing trees that never overlap? [closed]
I have been pondering the Collatz conjecture recently as a mental exercise, and have run into a problem that has to do with proving that an iteratively growing tree of odd positive integers will ...
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A chaotic function related to the $3x+1$ problem? (Li-Yorke and the Collatz problem)
Let $ x $ be an infinite binary string. Define the function $ f(x) $ mapping $ x $ to the Cantor set of $ I = [0,1] $ as:
$$
f(x) = \sum_{n=0}^{\infty} \frac{2 x_n}{3^{n+1}}
$$
where $ x_n $ are the ...
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Collatz conjecture and prime numbers [closed]
With the intention of understanding how prime numbers contribute to the numerical results we get when we perform any possible numerical calculation (also on real numbers), since they are those natural ...
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What's The Minimum Number Of Prime Factors Needed To Replace "3x+1" With Any Linear ("mx+b") Function And Still Work Like The Collatz Conjecture?
Apologies; I know there are a few assumptions used to pose this question, namely:
1): That yes, any mx+b function can work like the infamous "3x+1," problem...
...Provided, that you give it ...
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For each integer $k,$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},$ s.t. for each $n,\ p_{n+1}=2p_n- 1$ or $p_{n+1} =2p_n+1?$
For each $k\in\mathbb{N},$ does there exist a $k-$tuple of primes,
$(p_n)_{n=1}^{k},\ $ such that for each $n,$ the following is
satisfied: $p_{n+1} = 2p_n- 1\ $ or $p_{n+1} = 2p_n + 1?$
If yes then ...
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Is the number 3 in the Collatz conjecture arbitrary?
One of the most famous conjectures in mathematics is the Collatz conjecture also known as $3n+1$ but my question is why we multiply the odd number by 3? I get that the conjecture probably wants to ...
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Does this mean that there are no non trivial cycles in the Collatz Conjecture?
I am a grade 9 student who is fascinated with the Collatz Conjecture. I tried to 'prove' that the $4-2-1-4$ cycle is the only cycle that exists. I am aware that there is a huge chance that this is ...
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Is this Collatz-like stochastic process almost surely bounded?
I recently asked a question about whether a stochastic process $X =(X_n)_n$ could exhibit two properties at the same time, namely that $P(\sup_n X_n < \infty) = 1$ while $E(X_n) \to \infty$ as $n\...
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If a collatz sequence doesn't converge, how many odd numbers can its subsequence have?
Let a Collatz sequence of a natural number $x$ as follow:
$$\{x\}: \begin{cases}
x_0=x\\
x_{n+1} = \begin{cases}
\frac{3x_n+1}{2} \text{ if $x_n$ is odd}\\
\frac{x_n}{2} \text{ if $x_n$ is even}
\end{...
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Two-player Variant of Collatz procedure? [closed]
Recently i got interested at a Relaxation of Collatz Conjecture. This goes just like the normal Collatz procedure, except when $x$ is even, you have the choice of applying either $x \rightarrow 3x+1$ ...
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Hybrid between $5x+1$ and $7x+1$ that is probably convergent
Both the $5x+1$ and $7x+1$ variant of the Collatz sequence are conjectured to have large number of divergent trajectory. Here, i combined the two. As always, when you encounter even $x$, you apply $x\...
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Could a sequence in the Collatz conjecture actually increase without bound?
If my understanding is correct, than the Collatz conjecture could only be false if there is at least two closed cycle in it or if there is a number which increases without bound.
$3x-1$
We know that ...
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Proof for simplified Collatz conjecture variant with $10^{10^{100}} + 1$ as increment
Given the function
$$
S(n) =
\begin{cases}
\frac{n}{2}, & \text{if $n$ is even} \\
n + 1, & \text{if $n$ is odd}
\end{cases}
$$
Prove the conjecture: for any positive integer $n$, if you apply ...
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Seeking Literature on Fibonacci-related Patterns in Sequence Operations
Hey fellow math enthusiasts!
Problem:
The number breaking machine only processes natural numbers. Even numbers are halved, odd numbers are reduced by $1$, e.g. $6\to3$; $5\to4$.
Now the result is put ...
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The four sequences on the Collatz map
$$
T(n) =
\begin{cases}
\frac{n}{2} & n \equiv 0 \pmod{2}\\\\[2ex]
\frac{3n + 1}{2} & n \equiv 1 \pmod{2}
\end{cases}
$$
Starting with an arbitrary positive integer $k$, you can build a map ...
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The Collatz sequence, $\xi$ records
Consider the $3n+1$ sequence.
Let be $\sigma(n)$ the Number of steps necessary to reach the maximum of the trajectory starting from an integer $n$.
Let $\tau(n)$ be the Number of steps necessary to ...
user1225681
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The most famous trajectory of $3x+1$ problem
I think that the most famous and beautiful trajectory of the $3x+1$ problem is without doubt that starting from $n=27$ and having a maximum at $9232$.
The thing that I find very beautiful is that:
$$...
user1225681
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5
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Where can i find a proof that the allowable dropping times on the collatz conjecture are $\lfloor1 + n \cdot \log_2(3)\rfloor$
In the shortcut collatz function
$$
T(x) =
\begin{cases}
\frac{x}{2} & \text{if } x \equiv 0 \pmod{2} \\[2ex]
\frac{3x + 1}{2} & \text{if } x \equiv 1 \pmod{2}
\end{cases}
$$
The dropping time ...
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Question on a Collatz-like problem
For a positive integer $n$ define $f(n)$ as:
\begin{equation}
f(n) =
\begin{cases} n/2, & \text{if $n$ is even}\\
n + \lfloor n^x \rfloor, & \text{if $n$ is odd.}
\end{cases}\...
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How many affine prime-quotient ultrafilters does a rational semiring have?
I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of ...
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What is the pattern in the modulus when applying collatz function to linear polynomials?
$C(n) = \frac{n}{2}$ if $n$ is even, or $3n + 1$ if $n$ is odd
This function is normally applied to constant numbers, but it can be used on linear polynomials $ax + b$.
There are 3 possibilities of ...
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Why are Fibonacci numbers on the Collatz conjecture function?
$C(x) = \frac{1}{2}x$ if $x$ is even, $3x + 1$ if $x$ is odd
I will use $E$ to refer to an even number, and $O$ to refer to an odd number
$C(E)$ may be $E$ or may be $O$, as an even number divided ...
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Rough average length for the hailstone sequence produced from $n$?
The hailstone sequence for a number $n$ has you repeatedly replacing it with $\frac{n}{2}$ (if even) or $3n + 1$ (if odd), until you reach 1, at which point the sequence stops. Let's call the length ...
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Shortcut Collatz function satisfies a particular functional equation. Has this approach been studied yet, and if so where are the reference articles?
Let $X = 2\Bbb{Z} + 1$ or $2 \Bbb{N} + 1$ where $0 \in \Bbb{N}$, this approach will probably play well with both forms. See Extending the Collatz function to larger domains.
Define the shortcut ...
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Why do prime numbers have higher "lower bounds" for the maximum value in a Collatz sequence compared to composites?
I was playing around with collatz sequence stuff recently, and I made a plot of seed values vs the maximum value the seed's collatz sequence will reach. Highlighting primes in a logarithmic graph ...
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A Special Case of Loops in Collatz Conjecture
I want to show that there can't be simple loops in the Collatz conjecture over positive numbers, where you began with an odd number n, then after applying $\frac{3n+1}{2}$ then you repeat the process ...
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Is this a possible part of the solution to the Collatz Conjecture
I'm not trying to prove the Collatz Conjecture, but I think I may have found a pattern that could help solve it. This looks at the number of times a number will go up and down before reaching 1. For ...
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Is there a proof concerning the Collatz Conjecture that all odd numbers (n) divisible by 3 eventually encounter a number divisible by 8?
I'm vaguely aware that people have proved that by starting with any number, and iterating through the Collatz algorithm one will eventually reach a number divisible by 4.
But I am looking for the ...
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Proof attempt for a weaker form of the Collatz Conjecture
I am kind of new to this problem and I tried solving it with open mind. Please don't be judgmental, this is what I got.
Let us assume, for the sake of contradiction, that the Collatz conjecture is ...
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Advantages to formulating the Collatz Conjecture without division?
Usually, the Collatz function is defined by $C(n)=3n+1$ if $n$ is odd, and $C(n)=\frac{n}{2}$ if $n$ is even. The famous conjecture states that some iterate of $C$ evaluated at $n$ is $1$, for any ...
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$3n+3$ conjecture
While working on the Collatz problem, I came across this answer. I understand everything except for one thing: that for $n$ odd running Collatz2($n$) is exactly like running Collatz($n+1$). I ...
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Function for a new math pattern that emerged while working on the Collatz conjecture
So, this is a follow up to my previous question on the same topic, and in this question, I used the same technique, only with a larger value. Here's the set below:
S no.
Resultant Value
1
227
2
...
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2
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Closed formula for distances relating to Collatz conjecture?
I've been messing with the collatz conjecture for a while now, and I've found that another way to prove it is through proving that for any number $n$ there is at least one $k$ ($n$ is any odd input ...
3
votes
1
answer
535
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Collatz stopping time curves
We define the Collatz stopping-time of an integer $n$ to be the number of iterations of the Collatz function on $n$ untill we reach $1$. A corollary of the Colatz conjecture is that this time is ...
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How to find a function for this set of numbers I found while working on the collatz conjecture?
So, I was looking at the Collatz conjecture, and I thought of trying to reverse engineer the patterns in a certain sense, forming branches and trees. I figured it our for Branch-1, the formula, but ...
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Collatz variant $7 x + 1$?
Let $n$ be a positive integer
Now define the collatz variant
if $n=2m$ divide by $2$ as often as possible.
if $n=3m$ divide by $3$ as often as possible.
if $n=5m$ divide by $5$ as often as possible.
...
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1
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Collatz conjecture - Why does it end and not go on to infinity? [closed]
I was messing around with the sequences of odd numbers in the Collatz conjecture, and (unsurprisingly) found a pattern. Basically, I was calculating the number of steps an odd number takes to reach an ...
3
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1
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How to make sense of this plot $x-\sum[(\text{oddsteps})\mod 3]$
I am trying to make sense of this graph $y=x-\sum\limits_1^x(i\mod 3)$ where $i$ is the number of odd steps for $x$ to reach $1$ in a Collatz sequence.
(plot of $x$ from $1$ to $9\cdot 10^6$)
The ...
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