Dynamics of the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even. Generalizations to $n \to 3n-1 $ or $ n \to 5n+1$ or even to $n \to pn+q$ . Other names are "$3x+1$-problem","syracuse problem". If you have a question, please be specific to your detail. MSE is not ...

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Collatz Conjecture, sufficient to show odd numbers reach $1$?

The famous conjecture: Let $$ f(n) = \begin{cases} n/2 & \quad \text{if } n \text{ is even}\\ 3n+1 & \quad \text{if } n \text{ is odd}\\ \end{cases} $$ The Collatz Conjecture ...
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Determining the Collatz Series as a Tree of $\forall\mathbb{N}$

I'm proposing a proof for the Collatz Conjecture; and should like to take answers in terms of validation or contradiction to the arguments proposed. The conjecture states, where; $$ T(n) = \...
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On a proof that “there are at least $F_n$ Collatz permutations of length $n$”.

Let $n, k \in \Bbb{N}$ and $F_n$ be the $n$th term of the Fibonacci sequence. Let $u$ be the map $x \to 3x+1$ and $d$ be the map $x \to \frac{x}{2}$. Let a type be a sequence of $u$'s and $d$'s. ...
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How to prove equally likely steps of the Hailstone sequence (collatz sequence)

Consider the condensed collatz conjecture if $x$ odd then $f(x)=(3x+1)/2$: if $x$ even $f(x) = x/2$: Continue until $x = 1$ or find an $x$ in the natural numbers that will not hit $1$. The equation ...
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Closed form of iterated function

It can be easily calculated that performing the operation $(3x+1)$ on a number m, k times yields the result $(3^k)(m)+(3^k -1)/2$ I want to calculate a closed formula for performing the related ...
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Don't know what this expanding periodic-ish function is

I plotted a function $c(x)$, which returns $3x + 1$ if $x$ is odd, and $x/2$ if $x$ is even. It's the Collatz conjecture. I get this interesting function. I don't know what it's called, so I can't ...
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Consequences of Collatz Conjecture being true

Collatz conjecture has been conjectured for a long time and I think there are some evidence showing that it should be true. Similar to $P \neq NP$ conjecture, is there some interesting ...
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Is this a valid statement that would imply the Collatz Conjecture?

Let $f$ denote the Collatz transformation: $f(x) = \left\{ \begin{array}{ll} {x\over 2} & \quad x\equiv 0 \mod 2 \\ 3x+1 & \quad x \equiv 1\mod 2 \end{...
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The longest known cycle length of generalized collatz $5x+1$ trajectory

The generalized collatz $5x+1$ trajectory, if $n$ is even then divide $n$ by $2$, and if $n$ odd then multiply $n$ by $5$ and then add $1$. For example if $n=3$, we have $3=>16=>8=>4=>2=&...
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How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
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Extension of the collatz function to $\mathbb{C}$

The 3x+1 map is give as $$f(x) = \begin{cases} \frac{3x+1}{2} & \text{ if x odd} \\ \frac{x}{2} & \text{ else} \end{cases}$$ with domain $\mathbb{N}.$ On this wikipedia article, I found ...
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Collatz Conjecture: Literature on Convergence

Does anyone know of a paper showing that if all n converge, they must converge to unity for n>0. Else, any literature related to convergence properties would be appreciated. Thanks, Jordan
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Collatz-type problems with known divergent trajectories

It is a well-known problem, known as Collatz problem, to determine whether iteratively applying the map $f(x)=\frac{x}{2}\text{ if $x$ is even and }f(x)=3x+1\text{ otherwise}$ on a positive integer $n$...
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Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
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119 views

Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map $m\...
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Collatz algorithm generalization try-out (Collatz k-algorithm)

(Text Updated 2015/09/16, please see edit comments for changes) Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of ...
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Collatz Patterns

I have seen documentation on the $4K+1$ pattern, but as of yet I have seen nothing on the $64K+35$ pattern or the $262144K+184471$ pattern. Is there anywhere I can read up on these? I created the ...
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Counterexample 3n + 1 problem (Collatz) Exponential and linear Diophantine equation

So, I have found a sufficient condition (not necessary) for finding a counterexample to the 3n + 1 problem, namely the existence of solutions for the following two-parameter family of Diophantine ...
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What facts are known about the hypothetical smallest divergent integer in the collatz conjecture?

If there is a divergent integer in the collatz conjecture then there must be a smallest divergent number by the WOP. We can observe some properties of this number such as it must be odd because if it ...
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Has it been proven that there is no closed form for the hailstone numbers?

I know none has been found, and there probably isn't one considering the effort people have put into it, but has it been proven? (for some reasonable definition of "closed form"). I'm mostly ...
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Tag systems to cyclic tag systems and turing completeness

Consider the 2-tag system Alphabet: {a,b,c} Production rules: a --> bc b --> a c --> aaa and stating words aaa...a halts. on ...
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Can Collatz's problem be used as a pseudo random prime sieve?

If you take the concept of $3x+1$, $\dfrac{x}{2}$ and starting at 2, create a tree. On the left nodes you apply the $3x+1$. On the right nodes, if the parent node is even apply the $\dfrac{x}{2}$. ...