Bounds for the 3x+1 problem using difference inequalities
Copyright policy )Bounds for the 3x+1 problem using difference inequalities
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.
21 pages latex, 3 figures
- AT&T Labs United States
- Brunel University London United Kingdom
11B83 Primary, \(3x+1\) problem, Mathematics - Number Theory, difference equations, Dynamical Systems (math.DS), Collatz problem, 11B83 Primary; 26A18, 32H50 Secondary, Special sequences and polynomials, FOS: Mathematics, Recurrences, Number Theory (math.NT), Mathematics - Dynamical Systems, 26A18, 32H50 Secondary, Iteration of real functions in one variable
11B83 Primary, \(3x+1\) problem, Mathematics - Number Theory, difference equations, Dynamical Systems (math.DS), Collatz problem, 11B83 Primary; 26A18, 32H50 Secondary, Special sequences and polynomials, FOS: Mathematics, Recurrences, Number Theory (math.NT), Mathematics - Dynamical Systems, 26A18, 32H50 Secondary, Iteration of real functions in one variable
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