12 problems

Convergence conjecture for $(10,12,8)_+$. Repeating this process indefinitely, every starting number eventually reaches $4$ after finitely many iterations.

For $p\geq0$, let $(d_p,\alpha_p,\beta_p)_+=(2^{p+1},3\cdot2^p,2^p)_+$ and let $T_p$ be its associated mapping. Diagonal generalized Collatz conjecture. If $p\geq0$ and $p\neq2$, t…

For a positive integer $n\geq1$, if $n$ is a multiple of $10$, replace it by $n/10$; otherwise, let $r$ be its last digit, replace it by $(12n+8r)/10$, and iterate this process. Mo…

Let a satisfying cycle determine its dual object in Euclidean space by interpreting its associated tuple and its bitwise rotations as vertices in $\mathbb{R}^n$. A vertex is said t…

Let a tuple $(y_1,\dots,y_n)$ be satisfying when it satisfies the Loosened Collatz Function, and let $(2,2,\ldots)$ denote the constant tuple with every entry equal to $2$. Zero-en…

Let $\mathbb{N}_1$ denote the positive integers, and let the Loosened Collatz Graph be the graph whose vertices and cycles are defined by the Loosened Collatz Function. Cycle-verte…

Let $m\in\mathbb{N}$, and let $\operatorname{Col}$ be the Collatz function defined by … Consider the sequence $\operatorname{Col}^n(m)$ for $n\geq1$. Generalized Collatz conjecture…

Let … Let ${\cal O}_1=\{1,2\}$. An eventual finite orbit is a finite orbit in which the iterates of an initial integer eventually remain. Collatz orbit conjecture. ${\cal O}_1$ is…

Let $E$ denote the set of even integers, let $k\in E$, and let $n$ be an integer. The periodicity conjecture. The $n$-orbit is periodic. The preceding analysis shows divergence for…

Let $T_R:\mathbb{Z}\to\mathbb{Z}$ be defined by … For $k\geq 0$, define $T_R^{0}(n)=n$ and $T_R^{k+1}(n)=T_R(T_R^{k}(n))$, and call $(T_R^{k}(n))_{k\geq 0}$ the $T_R$-trajectory of…

Garner's conjecture. Every pair of corresponding stems is of the form $s_i$ and $s_i'$ for some $i$; equivalently, any pair of consecutive integers of the same height has parity ve…

Let $b$ and $m$ be the parameters defining the generalized Collatz map, and let $(S_k)$ be the sequence obtained by iterating this map from an arbitrary positive integer. Generaliz…