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

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The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$. We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
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I was wondering if it is known that the 3n+1 Collatz conjecture could be reframed as a statement about the set of solutions to a particular equation formulated as the sum of residues. This is ...
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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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@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{...
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Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows: Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
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Given a Dirichlet character $\chi$ modulo $8$ we consider the map $\mu(x)=x/2$ if $x$ is even and $\mu(x)=(3x+\chi(x))/2$ otherwise. (The corresponding map for $\chi$ the trivial Dirichlet character ...
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Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function: $$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$ We ...
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This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here. We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
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$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...
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As can be seen in the paper by Terras and Lagarias, it is possible to describe the results of the first $k$ iterations of the $3\cdot x+1$ problem by the parity vector $v(n)$. If $s^{(i)}(n)$ are the ...
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In their 2009 paper, Lagarias and Bernstein discuss the conjugacy map which is the unique homeomorphism $T$ on the $2-$adic field $\Bbb Q_2$ which fixes $0\mapsto0$ and topologically conjugates $x\...
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We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 ...
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As you all know, the Collatz conjecture claims that any positive integer will eventrually be reduced to 1 by appllying the sequence $n_{i+1} = x*n_{i} + 1$, when $n_{i}$ is odd, and $n_{i+1} = n_{i} / ...
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Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$. Each number $n$ represents a game played by left $L$ and right $R$: $$n = \{L_n | R_n \}$$ The rules ...
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Consider the sequence defined as follows: Start with a number N. Compute the prime factors of N with multiplicity, and add 1. Then, sum this list together to get N'. Iterate this procedure until you ...
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Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
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I have already asked on math.stackexchange if you think the question is off topic I can delete it. I'm trying to complete the following graph but I'm not sure if I can complete it without getting an ...
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I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...
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Does anyone ever write the Collatz conjecture as a single algebraic, recursive sequence? For example, a crude version might be: $$ g(n+1)=\delta _{1,g(n)}+(1-\delta _{1,g(n)})*\left(\left(\frac{cos(\...
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Given $a_0$ be an positive integer, define $$ a_{n+1} = \begin{cases} 8a_n, & \text{if $a_n$ is odd} \\ \lfloor a_n/3\rfloor, & \text{if $a_n$ is even} \end{cases}$$ Now form the sequence $(...
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There are all kinds of execution variants to the collatz conjecture for when hitting an odd number: $3n+1$ or $3n+3^a$ or $1.5n + 0.5$ or $1.5n + 1.5$... . The assumption is: proving any of them will ...
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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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In the Collatz iteration, it is known that the behavior of numbers congruent to 5 mod 8 exhibits interesting structure under modular transitions. Specifically, if we consider a sequence of 16,384 ...
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