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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 .

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