# 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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### Would proving that every positive integer > 1 gets mapped to 1 and only 1 unique point in the 2D plane be useful for proving Collatz Conjecture? [closed]

Before diving into the topic of the Collatz Conjecture, I'd like to briefly share my background, not as a means of boasting, but to provide some context to my perspective. I studied computer science ...
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### New Questions About Collatz Conjecture // Q1: Is there a cluster {o, L} for any odd number? [closed]

Odd Transformation Sequences (OTS) and Their Encoding in the Context of the Collatz Conjecture. Introduction The Collatz Conjecture, a longstanding problem in number theory, can be reframed through ...
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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

$\DeclareMathOperator\Orb{Orb}\newcommand\abs{\lvert#1\rvert}$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 Collatz conjecture known to be true for specific 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 ...
1 vote
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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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### Undecidable easy arithmetical statement

Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some ...
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### Are there infinitely many solutions of $2^k=3^z-1$ with $k,z \in \mathbb{N}$? [duplicate]

This question arose as an attempt to answer the following question Relaxed Collatz 3x+1 conjecture. I wanted to show that there is a solution of the equation $2^{k}=3^{z}(2n+1)-1$ for each $n\geq 2$, ... 1 vote
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### A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?

Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
Consider the function $f$ on the prime numbers defined by $$f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle (3,7,5,11,23,...