Bounds for the 3x+1 Problem using Difference Inequalities

Details Bounds for the 3x+1 Problem using Difference Inequalities Bounds for the 3x+1 Problem using Difference Inequalities

2002-04-30

arxiv.org/abs/math/0205002v1

Acta Arithmetica 109 (2003), 237--258.

Mathematics

Number Theory

21 pages latex, 3 figures

Scientific paper

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 x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.

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