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Bounds for the 3x+1 Problem using Difference Inequalities

Ilia Krasikov·Jeffrey C. Lagarias
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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 x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.

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Ilia Krasikov, Jeffrey C. Lagarias. 2002-04-30. Bounds for the 3x+1 Problem using Difference Inequalities. https://doi.org/10.4064/aa109-3-4

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