Questions tagged [collatz-conjecture]
The Collatz Conjecture, also known as the 3n+1 conjecture, is a famous open problem named after Lothar Collatz.
71
questions
13
votes
3
answers
2k
views
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 ...
17
votes
2
answers
1k
views
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}
\...
11
votes
1
answer
1k
views
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}$...
59
votes
3
answers
5k
views
Relaxed Collatz 3x+1 conjecture
The Collatz $3x+1$ conjecture claims that any positive integer can eventually be reduced to $1$ by iterative application of the maps $x \mapsto 3x+1$ whenever $x$ is odd and $x \mapsto x/2$ whenever $...
14
votes
1
answer
413
views
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 ...
9
votes
1
answer
892
views
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 ...
47
votes
1
answer
2k
views
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 ...
6
votes
0
answers
427
views
Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
22
votes
3
answers
2k
views
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:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
-1
votes
1
answer
371
views
Collatz property implying infinite "fall below" trajectories, is it known?
(this was discovered analyzing Collatz empirically.)
a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value.
consider a ...
5
votes
2
answers
2k
views
3n+1 problem and cycles
Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). ...
3
votes
0
answers
172
views
Largest permutation groups without "non-mixing" subgroups
We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives
if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit
representatives has natural density 0 ...
7
votes
1
answer
4k
views
Beyond Collatz: A $5n+1$ conjecture? [closed]
Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
1
vote
1
answer
236
views
Group with 2 orbits on the nonnegative integers -- description of the orbits
Definition: 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)}...
19
votes
3
answers
1k
views
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 ...
8
votes
1
answer
2k
views
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 $n=10^8$
(...
-3
votes
1
answer
673
views
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 ...
0
votes
1
answer
1k
views
Implication for cycles (of some length $m$) in Collatz-type problems: typical ratio between largest and smallest element?
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.
...
10
votes
2
answers
4k
views
Larger cycle than 4, 2, 1 in Collatz iteration?
(Here I discuss the Collatz problem only for positive integers.)
It is possible, by computation, to find all cycles in the Collatz iteration of a fixed length.
It is clear that an increase must be ...
-1
votes
1
answer
711
views
Collatz related question [closed]
Howdy,
Not sure this will be entirely clear, but when considering the relationship between a start value n in the Collatz algorithm and the length of the sequence generated by n, is there a function ...
33
votes
1
answer
3k
views
Collatz conjecture for numbers of th form $2^n +1$
Everybody has heard of the Collatz conjecture and it is a nice programming exercise to write a function, that calculates for a given number $n$ the number of iterations it takes until one reaches $1$. ...