Bounds for the 3x+1 Problem using Difference Inequalities
Ilia Krasikov, Jeffrey C. Lagarias
Abstract
We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x0.84 of the integers below x contain 1 in their forward orbit under the 3x+1 map.
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