Bounds for the 3x+1 problem using difference inequalities
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.
- Publication:
-
Acta Arithmetica
- Pub Date:
- 2003
- DOI:
- 10.4064/aa109-3-4
- arXiv:
- arXiv:math/0205002
- Bibcode:
- 2003AcAri.109..237K
- Keywords:
-
- Mathematics - Number Theory;
- Mathematics - Dynamical Systems;
- 11B83 Primary;
- 26A18;
- 32H50 Secondary
- E-Print:
- 21 pages latex, 3 figures