The \(3n+1\)-conjecture (Q2878187)

This paper is the text of lectures given during the `Vakantiecursus' (a mathematical summer course) organized by `het Platform Wiskunde Nederland' (platform for mathematics in the Netherlands).NEWLINENEWLINEIt gives a very nice introduction into the \(3n+1\)-conjecture from number theory and an overview of the developments since its formulation by Lothar Collatz in 1937. The very simple formulation uses the \textit{iteration function} defined for positive integers: NEWLINE\[NEWLINET(n)=\left\{ \begin{matrix} {1\over 2}n & \text{if }n\text{ even}, \cr {1\over 2}(3n+1) & \text{ if }n\text{ odd.}\cr \end{matrix}\right.NEWLINE\]NEWLINE The path \(\{T^k(n)\}_{k\geq 0}\) is then called \textit{convergent} if \(T^m(n)=n\) for an \(m\) (the \textit{cycle} \(\{n,T(n),\ldots,T^{m-1}(n)\}\) then repeats itself) and \textit{divergent} otherwise. The pair \(\{1,2\}\) is called the trivial cycleNEWLINENEWLINEThe conjecture is then formulated as:NEWLINENEWLINE\textbf{The \(3n+1\)-cycle conjecture}. The iterated \(T\)-function does not have non-trivial cycle.NEWLINENEWLINE\textbf{The \(3n+1\)-convergence conjecture}. The iterated \(T\)-function does not have any divergent path.NEWLINENEWLINEThe author gives a historical overview and formulates the situation in graph-theory language. Finally three equivalent formualations of the original \(3n+1\)-conjecture are given (two in matrix-language and one in functional analysis language).