
Contents
Next: Fractional parts of.
Up: Generalizations of problem.
Previous: Algorithmic decidability questions.
![[Annotate]](/organics/icons/sannotate.gif)
![[Shownotes]](../gif/annotate/sshow-131.gif)
Several authors have investigated the range of validity of the result that
has a finite stopping time for almost all integers n by considering more
general classes of periodicity linear functions.
One such class
consists of all functions
which
are given by
where m and d are positive integers with
and
,
is a fixed set of residue class representatives of the nonzero residue
classes
.
The 3x+1 function T is in the class
.
H. Möller [54] completely characterized the
functions
in the set
which have a finite
stopping time for almost all integers n.
He showed they are exactly those functions for which
E. Heppner [41] proved the following quantitative version of this result,
thereby generalizing Theorem D.
Theorem Q
(Heppner).
Let
be a function in the class
.
(i) If
, then there exist real numbers
such that for
we have
and
as
.
(ii)
If
, then there exist real numbers
such that for
we have
and
as
.
J.-P. Allouche [1]
has further sharpened Theorem Q and Matthews and Watts [51], [52] have
extended it to a larger class of functions.
It is a measure of the difficulty of problems in this area that even the
following apparently weak conjecture is unsolved.
EXISTENCE CONJECTURE.
Let U be any function in the class
.
Then:
(i) U has at least one purely periodic trajectory if
;
(ii)
U has at least one divergent trajectory if
.

Contents
Next: Fractional parts of.
Up: Generalizations of problem.
Previous: Algorithmic decidability questions.