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For questions about the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even, and its generalizations.
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Can someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?
Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '... number-theory modular-arithmetic dynamical-systems collatz-conjecture music-theory- 113
Is it possible to find closed form of the Collatz conjecture?
So recently I was thinking about converting the recurrent definition of Collatz conjecture into a closed-form expression, which would map any $n,\space n\in\mathbb{N}$ to the $n^{th}$ iteration of the ... discrete-mathematics collatz-conjecture- 19
Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?
Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ... number-theory recreational-mathematics collatz-conjecture- 12.5k
Why does this cycle of 44 show up in the Collatz Conjecture?
Consider this function: $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{... number-theory modular-arithmetic dynamical-systems polar-coordinates collatz-conjecture- 113
Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem
It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ... combinatorics number-theory reference-request collatz-conjecture- 9,935
Is there a name for this function or a concept similar to it?
I'm wondering if anyone has heard or seen a function that looks or behaves like this one. $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{... elementary-number-theory dynamical-systems exponentiation collatz-conjecture- 113
Construction of Collatz tree from a matrix
Let us build the Collatz tree from a convenient matrix. The matrix has the odd integers at the bottom. Every element in the rows above is the double of the element below. The result is the following. $... collatz-conjecture- 4,924
A typo in an implementation of the Collatz algorithm made it loop infinitely
This question is kind of related to programming but my question only relates to math in a theoretical sense. I was trying to implement a Collatz conjecture algorithm that calculates the steps it needs ... terminology collatz-conjecture- 101
Is there an explanation for clustering of total stopping times in Collatz sequences?
I assume knowledge of the Collatz conjecture. Here I'm looking only at total stopping times $t(n)$, and mostly will drop the word "total." Just looking at relatively simple graphs of ... number-theory collatz-conjecture- 1,268
Collatz Conjecture and non-trivial cycles
Consider $$n \rightarrow ... \rightarrow an+b$$ to be a sequence of natural numbers to which we apply $3n+1$ or $\frac{n}{2}$ operations. This sequence is called a cycle, if and only if $an+b=2n$. (... number-theory collatz-conjecture exponential-diophantine-equations- 339
Has anyone discovered this Collatz Conjecture pattern?
I noticed that there is a linear increase of +1 in the stopping times of a sequence of numbers with the seeds being the sum of the previous number in the sequence added to itself. I tried this on ... collatz-conjecture- 1
For any composition sequence $s$ of maps $h(X)=X/2, \ f(X)=(3X + 1)/2$, there exists an integer $X$ such that its Collatz sequence contains $s$
Let $h(X) = X/2$, and $f(X) = (3X + 1)/2$. Then clearly every iteration $g^i(X), X \in \Bbb{Z}$ the Collatz mapping $$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \ X = 1\pmod 2 \end{... elementary-number-theory solution-verification conjectures collatz-conjecture iterated-function-system- 19.2k
Terras lemma proof
For any number N the parity vector v(N) is defined as $v_i(N) = S_i \pmod 2$ If N is a positive integer of the form $a\cdot 2^k + b (b < 2^k)$ then the first k elements of the parity vector are ... number-theory collatz-conjecture- 1
My proof that there exist no odd 2-cycles for the accelerated Collatz function other than $(1,1)$. Do you have a link to a historical proof?
Let $f(x) = \dfrac{3x + 1}{2^{\nu_2(3x + 1)}}$. Clearly the $(1,2,4)$ cycle from the original Collatz map disappears under this context. So the Collatz conjecture-equivalent goal with $f$ is that ... linear-algebra elementary-number-theory soft-question collatz-conjecture- 19.2k
If there are $k_1, k_2, \dots, k_n$ divisions by $2$ in a Collatz cycle, then $k_1 + \dots + k_n \geq n$, but can we get a greater lower bound?
Let $f(x) = |3x + 1|_2(3x + 1)$ be the accelerated Collatz function, where $|3x + 1| = 2^{-\nu_2(3x + 1)}$ is the $2$-adic absolute value. Clearly for all $x$ odd we have $\nu_2(3x + 1) \geq 1$ so ... elementary-number-theory reference-request upper-lower-bounds collatz-conjecture open-problem- 19.2k
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