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Mathematics > Number Theory

arXiv:math/0205002 (math)
[Submitted on 30 Apr 2002]

Title:Bounds for the 3x+1 Problem using Difference Inequalities

Authors: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.
Comments: 21 pages latex, 3 figures
Subjects: Number Theory (math.NT); Dynamical Systems (math.DS)
MSC classes: 11B83 Primary; 26A18, 32H50 Secondary
Cite as: arXiv:math/0205002 [math.NT]
  (or arXiv:math/0205002v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.math/0205002
Journal reference: Acta Arithmetica 109 (2003), 237--258.
Related DOI: https://doi.org/10.4064/aa109-3-4

Submission history

From: Jeffrey C. Lagarias [view email]
[v1] Tue, 30 Apr 2002 21:57:00 UTC (26 KB)
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