The \(3x+1\) problem: New lower bounds on nontrivial cycle lengths
The ``Collatz''-problem (or ``\(3x+1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n\in\mathbb{N}\) there exists a \(k\) with \(T^{(k)}(n)=1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\). This conjecture has been verified with a computer at least up to \(n=2^{40}\). Given \(n\in\mathbb{N}\), a trajectory of \(n\) is defined as the set of iterates \(\Omega(n):= \{n,T(n), T^{(2)}(n),\dots\}\). A trajectory \(\Omega\) will be called a cycle (of length \(k\)) if \(T^{(k)}(x)=x\) for all \(x\in \Omega\), where \(k= \text{Card }\Omega\). For example, \(\Omega(1)= \{1,2\}\) is a cycle of length 2, called the trivial cycle. Now the ``\(3x+1\)''-conjecture reads as \(1\in\Omega(n)\) for all \(n\in\mathbb{N}\) and implies that \(\Omega(1)\) should be the only cycle of \(T\). Various authors have observed that nontrivial cycles of \(T\) (if any) must be very big. In this paper the author proves that for any nontrivial cycle of \(T\) with \(\text{Min} \Omega>2^{40}\) holds the equation \[ \text{Card }\Omega= 301 994a+ 17 087 915b+ 85 137 581c, \] where \(a\), \(b\), \(c\) are nonnegative numbers, \(b>0\), and \(ac=0\). In particular, the smallest admissible values for \(\text{Card }\Omega\) are 17 087 915, 17 389 909, 17 691 903, and so on. In his proof the author uses classical facts about continued fractions to obtain a sufficient sharp one-sided diophantine approximation of \(\log_ 2(3)\).
one-sided diophantine approximation of \(\log_ 2(3)\)
scientific article; zbMATH DE number 417460
scientific article