A simple (inductive) proof for the non-existence of 2-cycles of the \(3x+1\) problem
When \(x\) is a positive integer, define \(T(x)=\frac{1}{2}(3x+1)\) if \(x\) is odd and \(T(x)=\frac{1}{2}x\) if \(x\) is even. It is conjectured that for every positive integer \(x\) the iterates of \(T\) eventually produce the cycle (1,2). A 2-cycle for the \(3x+1\) problem is a cycle having two odd local minima \(x_0\) and \(x_1\) of the shape \(x_i=a_i2^{k_i}-1\). The author provides a new proof that there are no 2-cycles other than the trivial one given by \((1,2,1,2)\). This is achieved by deriving upper and lower bounds for the quantity \(a_0a_1\) associated to a non-trivial 2-cycle and then applying work of \textit{B. M. M. de Weger} [Algorithms for Diophantine equations. CWI Tract, vol.\ 65, Centre for Mathematics and Computer Science, Amsterdam (1989; Zbl 0687.10013)] on linear forms in logarithms to deduce that the bounds are contradictory.
scientific article; zbMATH DE number 5129741
scientific article