The 3x+1 Semigroup
Abstract
The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for nonnegative 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2004
 DOI:
 10.48550/arXiv.math/0411140
 arXiv:
 arXiv:math/0411140
 Bibcode:
 2004math.....11140A
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Dynamical Systems;
 11B83
 EPrint:
 16 pages, latex