# Questions tagged [collatz-conjecture]

The Collatz Conjecture, also known as the 3n+1 conjecture, is a famous open problem named after Lothar Collatz.

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### The two Collatz-maps associated to characters modulo 8

Given a Dirichlet character $\chi$ modulo $8$ we consider the map $\mu(x)=x/2$ if $x$ is even and $\mu(x)=(3x+\chi(x))/2$ otherwise. (The corresponding map for $\chi$ the trivial Dirichlet character ...
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### Borderline Collatz-like problems

The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$. We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
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### Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern. We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
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### Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...
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### Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (in June 2016): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \...
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### Some Questions on the Collatz conjecture (reexpressed as "equivalence relation")

The set of all positive whole numbers is denoted by $\mathbb{N}_+$. Let $f\colon\ \mathbb{N}_+\to\mathbb{N}_+:n\mapsto \begin{cases}\frac{n}{2}&\text{$n$even}\\3n+1&\text{$n$odd}\end{cases}$...
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### A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:  f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
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### Collatz property implying infinite "fall below" trajectories, is it known?

(this was discovered analyzing Collatz empirically.) a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value. consider a ...
We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 ...