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.

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Can someone give some constructive criticism for this disproof of the Collatz conjecture [closed]

My name is JaMarkus Brown. I'm from Fielder Elementary. I told my teacher about my work, and she sent me here. A couple days ago, I was thinking about the Collatz conjecture. So I made a little ...
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Gaps between elements of a collatz sequence [closed]

Let $C: \mathbb{N} \rightarrow \mathbb{N}$ be your typical collatz map (i.e. $3N+1$ for odd N, $N/2$ for even N). My question is about $f(N) = min_{j≥1} \{|N-C^{(j)}(N)|\}$. Now, assuming Collatz ...
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Is this conjecture true? Is it known?

I discover a variant of the Collatz Conjecture—the Caveman Conjecture—and it goes as follows: $$ \begin{cases} n \to 2n + 2 & \text{ if $n$ is odd}\\ n \to \frac{n}{2} & \text{ if $n$ is even}\...
caveman's user avatar
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The Mutated Collatz Sequence: At each step, there's probability $\varepsilon$ of adding $1$

I created a new sequence (mutated collatz sequence), where the rules goes just like the normal Collatz Sequence but after every iteration, there's a probability $\varepsilon$ that we add 1. So for a ...
Py Py's user avatar
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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 ...
RBen's user avatar
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Random Walks and Optimal Return Paths in a Collatz-Inspired Tree of Odd Integers

In a Collatz-inspired tree comprising solely odd integers, an 'ant' simulates random walks utilizing three operations and their inverses: $ D $: $ i' = (i-1)/4 $, requiring $ i' $ to be odd $ V $: $ ...
PMF's user avatar
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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 ...
Wagner Martins's user avatar
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1 answer
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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 ...
Appollonius's user avatar
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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 ...
Wagner Martins's user avatar
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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 ...
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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: $$...
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5 answers
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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 ...
Wagner Martins's user avatar
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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 ...
it's a hire car baby's user avatar
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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 ...
Wagner Martins's user avatar
2 votes
1 answer
199 views

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 ...
Wagner Martins's user avatar
2 votes
1 answer
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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 ...
Daniel Donnelly's user avatar
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1 answer
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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 ...
Desmond's user avatar
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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 ...
Berny's user avatar
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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 ...
MetaStack's user avatar
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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 ...
Yoav Alhindi's user avatar
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1 answer
191 views

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 ...
Louis Rubin's user avatar
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2 answers
252 views

$3n+3$ 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 ...
Tsar Asterov XVII's user avatar
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2 answers
250 views

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 ...
Michael Iacovacci's user avatar
2 votes
1 answer
287 views

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 ...
Rd Basha's user avatar
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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 ...
Tsar Asterov XVII's user avatar
5 votes
1 answer
444 views

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. ...
mick's user avatar
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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 ...
Samuel Dupont's user avatar
4 votes
1 answer
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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 ...
user489810's user avatar
2 votes
2 answers
286 views

Collatz conjecture generalization's further discussion

This question is connected with another one found in the following link: Collatz conjecture generalization Consider the following algorithm: For a given prime $p$, take any integer and, if it is ...
Igor Paulino's user avatar
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Collatz conjecture generalization

I'm studying the collatz conjecture and I arrived at the following generalization: Given any prime p, take any integer n and apply the following process: If n is divisible by any prime smaller than p, ...
Igor Paulino's user avatar
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2 answers
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How do I solve a system of equations containing a modulus function?

I'm trying to solve this system of equations for $a$: $$\frac{3n+1}{2^{a-1}} \mod 2 = 1$$ $$n \mod 2 = 1$$ for any odd input $n$. I know that there is only $1$ solution for every possible $n$ value. ...
user avatar
1 vote
1 answer
340 views

Proof verification: $3n+3^k$ problem equivalent to the Collatz problem

The Collatz conjecture asserts that a sequence defined by repeatedly applying the function \begin{align} T_0(n) &= \begin{cases} (3n + 1) / 2, & \text{for odd $n$,}\\ n / 2, &...
DaBler's user avatar
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Considerations on the sequences of the 3n + 1 problem

Starting from an odd number $n$, example analyzing the sequence for $\quad n=57$ $57\quad\rightarrow 172\rightarrow 86\rightarrow 43\rightarrow 130\rightarrow 65\rightarrow 196\rightarrow 98\...
user140242's user avatar
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Do the self-similarities of a fractal function govern its convergence properties?

Let the graph of a function take a fractal form, such as the following representation of the Collatz conjecture topologically conjugated to the interval $[\frac12,1)\to[\frac12,1)$ In this example, ...
it's a hire car baby's user avatar
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Collatz Conjecture Interpretation?

Another way to interpret the Collatz conjecture is to say that "playing the game" will always arrive at 2^n. The natural numbers can be rearranged into "rails" of each odd number ...
Jack's user avatar
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1 answer
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An argument that the value of m in a Collatz conjecture 2-cycle is subject to the same constraints as for a 1-cycle.

Question: Is the reasoning of the following argument valid? Objective: To derive a constraint on the permissible number of total divisions by 2, here referred to as $m$, for a Collatz 2-cycle ...
Joe B.'s user avatar
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2 answers
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General form for the k-th next odd number the Collatz conjecture

Let's consider only the odd positive integers in the Collatz conjecture. If the conjecture is true, they'd form a directed graph pointing to 1, which points to itself. The next odd number in the graph,...
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1 answer
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What is the minimum nontrivial cycle length in the Collatz problem?

Recently, I have read the article (Eliahou, 1993). The author shows that any nontrivial cycle must contain at least 17 087 915 elements, assuming that the Collatz problem has been verified up to $2^{...
DaBler's user avatar
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Prove that $f(x) = -1-3x \pmod{2^e}$ cycles with period $2^e$

This question is inspired by the Collatz-like "Binary Complement Sequences," as discussed here and here and here on seq-fan. For a given exponent $e$, let $f$ be defined as $f(x) = -1-3x \...
Quuxplusone's user avatar
5 votes
1 answer
672 views

Collatz Conjecture Inquiry

I recently decided to look into the problem for fun, and I came across a pathway to a proof, and, though it's rather long, I was wondering if it has been investigated before. So, for the Collatz ...
M.L's user avatar
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0 answers
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A group presentation given by the Collatz sequence of a fixed $n$. Has this been studied before?

This is a reference-request question. The Set Up: Fix $n\in\Bbb N$. Suppose $$C(x)=\begin{cases} x/2&:x\text{ is even},\\ 3x+1&:x\text{ is odd}. \end{cases}$$ Let $m\in\Bbb N\cup\{0\}$. Denote ...
Shaun's user avatar
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Why do the odd numbers in a Collatz trajectory hit so many Primes?

I implemented the Collatz procedure in C++ and simulated all numbers up to n (I did it up to 100000). The program counted the total number of primes and the total number of composites hit (only out of ...
Finn's user avatar
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8 votes
1 answer
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Is there a way to predict periodic behavior in Collatz conjecture 1-cycles?

This is a recreational math question. If I use the equation for the smallest element in a 1-cycle,$$X_1 = \frac{3^n-2^n}{2^m-3^n}$$ with $n*ln_23 < m < n*ln_23 +1$, peak values of this function ...
Joe B.'s user avatar
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7 votes
2 answers
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Collatz-like problem involving prime factors

Unfortunately I am not well-versed in LaTeX so I will try my best to keep this looking presentable. As an overview, I was investigating a variation of the Collatz conjecture: Define $f(1) = 1$ Then, ...
B Kosta's user avatar
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1 answer
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A question about mathematical reasoning regarding observations in Collatz conjecture 1-cycles

This is a question about the use of empirical observations to guide strategies in constructing mathematical proofs. It uses prior discussions of 1-cycles in the Collatz conjecture as a model. In a ...
Joe B.'s user avatar
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5 votes
1 answer
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Deriving a better approximation for the sum of steps in the Collatz conjecture

Let $S(n)$ be the number of steps required for $n$ to reach $1$ in the 3n+1 problem (A006577). As showed in other posts, $S(n)$ is locally random, but a local average/estimate $s(n)$ can be calculated ...
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