# Tagged Questions

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

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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 (over the past week): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{...
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### Some Questions on the Collatz conjecture

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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### Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ ...
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### A Collatz-like question about permutations

An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting. Consider all permutations $\pi$ on the natural numbers such that ...
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### Residue class sufficiency sets for the Collatz conjecture

I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of ...
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### Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
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### A curious sequence of rationals: finite or infinite?

Consider the following function repeatedly applied to a rational $r = a/b$ in lowest terms: $f(a/b) = (a b) / (a + b - 1)$. So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$. I am wondering if it is ...
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### Collatz stopping-time and Poisson distribution, and connection to other problems?

I read many threads about Collatz here - so don't worry, this is no attempt to any proof, just asking about a curious fact: This graph gives the stopping-time of Collatz sequences up to $n=10^8$ (...
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### Point me to an attempt to Proove Collatz Conjecture by Substitution and Factor analysis? [closed]

Summary of Question: Where can I find a discussion of attempting to prove the Collatz Conjecture via substitution and abstract examination? I've done a lot of reading on the problem, including ...
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### Implication for m-cycles in Collatz-type problems.

Background Consider Collatz-type problems of the form $an + 1$, where $a > 2$ is a positive, odd integer (e.g., $3n + 1$, $5n +1$, $7n + 1$, etc.). For convenience, automatically divide by two. ...