Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three and add

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Define $\operatorname{Col}:\mathbb N\to\mathbb N$ by \[\operatorname{Col}(n)=\begin{cases}n/2,&n\text{ even},\\3n+1,&n\text{ odd}.\end{cases}\] For $j\ge0$, $\operatorname{Col}^j$ is the $j$-fold iterate, with $\operatorname{Col}^0(n)=n$.

Statement recorded in the supplied source.

Does every $n\in\mathbb N$ have an integer $j\ge0$ for which $\operatorname{Col}^j(n)=1$? [S2]

Short English statement

Does the repeated halve-or-triple-and-add-one process reach one from every positive starting integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100049 (AMR-030-0049). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Terence Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum of Mathematics, Pi 10 (2022), e12, Section1.1, definition of the Collatz map and Conjecture1.1, page1. https://arxiv.org/abs/1909.03562.