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

A 2-cycle of the 3x+1 problem has two local odd minima x0 and x1 with xi=ai2ki−1. Such a cycle exists if and only if an integer solution exists of a diophantine system of equations in the coefficients ai. We derive a numerical lower bound for a0dot operatora1, based on Steinerʹs proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a0dot operatora1 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.