A simple (inductive) proof for the non-existence of 2-cycles of the [formula omitted] problem

Abstract

A 2-cycle of the 3 x + 1 problem has two local odd minima x 0 and x 1 with x i = a i 2 k i − 1 . Such a cycle exists if and only if an integer solution exists of a diophantine system of equations in the coefficients a i . We derive a numerical lower bound for a 0 ⋅ a 1 , based on Steiner's proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a 0 ⋅ a 1 as a function of K and L (the number of odd and even numbers in the cycle). We apply a result of de Weger on linear logarithmic forms to show that these lower and upper bounds are contrary. The proof does not use exterior lower bounds for numbers in a cycle and for the cycle length.