A simple (inductive) proof for the non-existence of 2-cycles of the 3x+1 problem

A 2-cycle of the 3x + 1 problem has two local odd minima x(0) and x(1) with x(1) = a(i)2(ki) - 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) center dot 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) center dot 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. (c) 2006 Elsevier Inc. All rights reserved.