29 problems

The 3x+1 conjecture. Every integer $n\geq 1$ eventually reaches $1$ under iteration of $T$.

Let $T:\mathbf{Z}\to\mathbf{Z}$ be the $3x+1$ function defined by … For $n\geq1$, write $T^{(k)}(n)$ for the $k$-fold iterate of $T$. The 3x+1 conjecture. For each $n\geq1$, some i…

Strong-admissibility conjecture for $(2^p+2^q,2^p+2^{q+1},2^p)_+$. The triplet is strongly admissible for all such $(p,q)$. If $(p,q)\notin\mathbb{E}$, it has order one and every…

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

The $3n+1$ conjecture. Starting with any $n\in\mathbb{N}$, there is a positive integer $k\in\mathbb{N}_0$ such that $C^{(k)}(n)=1$.

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 integers $p\geq q\geq0$, set … and let $T_{p,q}$ be the associated mapping. A triplet is strongly admissible when it has finitely many cycles and no divergent trajectory. Gener…

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 $C:\mathbb{N}\to\mathbb{N}$ be the Collatz operator, with $C(n)=n/2$ for even $n$ and $C(n)=(3n+1)/2$ for odd $n$. A cycle is a finite periodic orbit of $C$. Uniqueness-of-cycl…

Let a Collatz permutation sequence be a trajectory under one of the permutation functions considered in the paper, and let the four known cycles be the cycles identified for these…

Let a dual object be the geometric object associated with a satisfying cycle, with vertices interpreted as points in $\mathbb{R}^n$. Let the line about which rotation is considered…

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…

Hasse–Prichett conjecture.

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…

Syracuse conjecture. For every $k\in I$, there exists an integer $n\geq 1$ such that

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…

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

Let $n$ be a positive integer, and define … Starting from $n$, repeatedly apply $f$ to obtain its orbit. The $3n+1$ conjecture. For every positive integer $n$, this orbit will alwa…

Let $n\geq 1$ and $d\geq 2$ be coprime integers, let $\mathcal R$ be a complete set of residues modulo $d$, and let $\varphi$ be the canonical surjection from the integers to…

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 $T:\mathbb{N}\to\mathbb{N}$ be the Collatz map … For $s\in\mathbb{N}$ with $s>1$, Collatz's stopping-time conjecture. There exists $k\in\mathbb{N}$ such that $T^k(s)<s$. This i…