The 3x+1 Semigroup
Number Theory
2007-05-23 v2 Dynamical Systems
Abstract
The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.
Keywords
Cite
@article{arxiv.math/0411140,
title = {The 3x+1 Semigroup},
author = {David Applegate and Jeffrey C. Lagarias},
journal= {arXiv preprint arXiv:math/0411140},
year = {2007}
}
Comments
16 pages, latex; minor changes