On generalized circuit of the Collatz conjecture
The ``Collatz-Conjecture'' (or ``\(3x+1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n \in \mathbb N\) (or \(n\in\mathbb Q)\) there exists an iteration \(k\) of the Collatz-function \(C(n)\) with \(C^{(k)}(n)=1\), where \(C(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\). A number \(m\) is called a cycle-number if some iteration of \(C\) will give \(C^{(k)}(m)=m\). It is well-known that the sequence of a cycle cannot be a circuit (i.e. the sequence consists of one strictly rising and one strictly descending part) until \(m=1\). In the note under review the author shows that a somewhat generalized cycle cannot exist among the natural numbers. His result depends essentially on a theorem from Diophantine approximation theory proved by G.~J. Wirsching.
scientific article; zbMATH DE number 5015948