# 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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### 7 for 5n+1 series diverges to infinity? [closed]

Write an odd integer in the "modified" binary form as $2^m - 1$ and apply 5n+1 (similar to applying the rules of original 3n + 1) to it: \begin{align} n &= 2^m - 1 \\ ...
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### Cocycles and the Collatz problem?

Let $T(n) = n+R(n)$, where $R(n) = -n/2$ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$. $R(n)$ is the Cantor ordering of the integers: https://oeis.org/A001057 In the Collatz problem, one is ...
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### Finding a strictly increasing Collatz sequence of arbitrary length [closed]

Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I ...
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### Can anyone recommend a reference where the collatz conjecture is viewed 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 ...
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### Polynomials, $3^x$ and the Collatz conjecture

$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open. Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
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### Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?

The Collatz or the $3n+1$ conjecture is open. Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
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### Can you explain this weird pattern in Collatz conjecture? [closed]

Extreme disproportion in the dispersion of "Digital Roots" of the highest numbers reach in the Collatz Conjecture. I calculated the Digital Root remainder mod 9 for the highest numbers ...
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### Summary of “Almost All Orbits of the Collatz Map Attain Almost Bounded Values”

Terence Tao's 2019 paper Almost all Orbits of the Collatz map attain almost bounded values" is pretty famous. However, it's also long and complicated. I think there are useful techniques to ...
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### Density of the Klarner-Rado Sequence

Consider the Klarner-Rado sequence OEIS A005658 defined by the rule: the sequence starts with 1, and if it contains $n$ it also contains $2n$, $3n+2$ and $6n+3$. According to R. Guy's popular article,...
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### How can I catalog these generalized Collatz problems?

The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$: $x + x + 1 \rightarrow x+x+x+1+1;$ $x + x \rightarrow x;$ Whenever a number matches the LHS ...
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### Extension of Coburn's theorem on isometry and Toeplitz algebra

$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...
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### A problem involving the inverse Collatz map

Let $C$ be the Collatz map on the natural numbers, defined by: $$C(n) := \begin{cases} n/2 & \text{if} \;n \;\text{even} \\ (3n+1)/2 & \text{if} \;n \;\text{odd} \end{cases}$$ The inverse ...
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### Identification of Invariant Sets for Discrete Dynamical Systems on the Positive Integers

Let $\phi:\mathbb{N}\times \mathbb{N}^+\rightarrow \mathbb{N}^+$ be a dynamical system on the positive integers. Suppose we refer to the orbit of a periodic point of $\phi$ as an invariant set of the ...
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