Contents
** Next:** Connections of the
**Up:** The 3x+1 problem.
** Previous:** Are there non-trivial

Several authors have observed that heuristic probabilistic arguments suggest
that no divergent trajectories occur.

###
DIVERGENT TRAJECTORIES CONJECTURE.

* The function has no divergent trajectories,
i.e., there exists no integer for which*
If a divergent trajectory exists, it
cannot be equidistributed .
Indeed if one defines

then it can be proved that the condition (2.30) implies that
Theorem F constrains the possible behavior of divergent trajectories.
Indeed, associated to any divergent trajectory is the
infinite set and for all
.
Since for all , Theorem F implies that

where .
Roughly speaking, (2.32) asserts that the elements of a divergent
trajectory cannot go to infinity ``too slowly.''

Contents
** Next:** Connections of the
**Up:** The 3x+1 problem.
** Previous:** Are there non-trivial