An E-Squence Approach to the 3x + 1 Problem

Abstract

For any odd positive integer x, define (x n ) n⩾0 and (a n ) n⩾1 of all 2 an is called the E-sequence of x. For example, the orbit and the E-sequence of 3 are (3, 5, 1, 1, . . .) and (1, 4, 2, 2, . . .), respectively.Given any sequence (a n ) n⩾1 of positive integers, if it is the E-sequence of the odd positive integer x, it is called to be convergent to x and, denoted by Ω − lim a n = x; if (a n ) n⩾1 is not the E-sequence of any odd positive integer, 2010 Mathematics Subject Classification. Primary 11A99; Secondary 11B83.

Computer-Based Validation of 3n+1 Hypothesis for Numbers 3n−1

“…Despite the fact that the formulation of the problem 3n+1 in terms of the game is plausible to everyone who knows basic mathematical operations, as far as we know the problem has not been solved. Up to this date there have been a couple of hundreds of publications which either directly or indirectly addressed the 3n+1 problem from various points of view (recent papers [10,11,12,13]).…”