Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4595201 | Journal of Number Theory | 2007 | 8 Pages |
Abstract
A 2-cycle of the 3x+1 problem has two local odd minima x0 and x1 with xi=aiki2−1. 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 a0⋅a1, based on Steiner's proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a0⋅a1 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory