Dynamics of the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even. Generalizations to $n \to 3n-1 $ or $ n \to 5n+1$ or even to $n \to pn+q$ . Other names are "$3x+1$-problem","syracuse problem". If you have a question, please be specific to your detail. MSE is not intended to check attempted proofs.

The Collatz Conjecture asserts that every integer, when iterated under a function taking odd $n$ to $3n+1$ and even $n$ to $\frac{n}2$, will eventually reach $1$. The questions around the conjecture are questions about the status of the $3n+1$ problem or that of its generalizations and about literature. Since it is an exponential diophantine problem which is iterated according to residues $ \pmod 2$, such questions often deal with modular arithmetic and subsequent problems. Also the general problem of approximation of powers of 2 to that of powers of 3 (in the original $3n+1$-formulation) occurs as determining ingredient.
Since it is an iterative problem, the topic of fixpoints and cyclic points is also relevant.
One might also consider the known strategies of proving; so questions concerned with such strategies of approaching the Collatz-problem occur sometimes. The Collatz conjecture is still unsolved and seemingly very difficult to attack.

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