WARNING - this page is both outdated and incorrect It is included only for anyone who might be interested in the history of the discovery that the state transition diagram for the Collatz itinerary is identical in the positive and negative domains. Otherwise it is certainly not worth taking the time to study it.
An article in Wikipedia extends the Collatz conjecture into the domain of negative integers, identifies 3 loops in the negative domain, and calls into question the proposition that the only loop in the Collatz trajectories among positive integers is the trivial 1-4-2-1 loop which terminates the iterations of (3n+1)/2^i (or various equivalents). The author uses residues modulo 6 of the iterates to discover these loops and questions whether additional loops in the positive integers may be still undiscovered.
In my work with the usual (3n+1)/2^i itinerary, I have made a state transition diagram employing the residues mod 3 and mod 8 (or equivalently mod 24) as the state descriptors. The diagram shows that the Collatz iterates are limited to well defined transitions and admits no possibility of a loop among the positive integers other than the trivial terminating one. As it turns out, it also develops left descent assemblies which provide the basis of the structure of the binary predecessor tree for the Collatz graph.
The use of state transition diagrams for an accurate overview of the transitions involved in any particular Collatz-like set of itineraries seems compelling since they give a relatively clear presentation. For example, the state diagram for (5n+1)/2^i, among others, is quite unlike that for (3n+1)/2^i and quickly and clearly discovers and elucidates the loops intrinsic to that analog. On the other hand, state transition diagrams for two analogs which do exhibit Collatz-like behavior affirm the absence of possible loops in their itineraries.
The state diagrams alluded to in the previous paragraphs all use residues mod 3 and mod 8 as the state descriptors. Certainly, the choice of state descriptors determines the appearance of the state transition diagram, so it is always possible to be misled by an unfortunate choice.
Constructing a state diagram requires listing out all possible
transitions for all possible states (column 1) of a given analog as is
done below. I have used values of the iterates mod 24 as state
descriptors, and, in an attempt at completeness, have done
the calculation for the first three members (column 2) of
each residue set. (But even so, a fourth member had to be
added to J's initial states to get the -91 to -17 step in the larger of
the loops to come.) The entries in column 3 of the table are
successors
in the forward iteration. Where a successor lies outside the range
of -1..-72, I have indicated the corresponding value of the
residue set member >-24 in column 4. Column 5 shows the set of
states whose successor is that of column 1 (i.e. the set of states
which are its predecessors). The states starred in column 3 are
reached
A -1 -1 A A<-A,B,E,F,I,J,L
-1[24] -25 -37 G
-49 -73 -1 A
--------------------------------
B -3 -1* A SOURCE (0[3])
-3[24] -27 -5* C
-51 -19* J
-75 -7* D
-99 -37* G
-123 -23* L
-315 -59* F
-363 -17* I
--------------------------------
C -5 -7 D C<-B,D,F,H,J,L
-5[24] -29 -43 J
--------------------------------
D -7 -5 C D<-B,C,F,G,J,K
-7[24] -31 -23 L
-55 -41 I
-79 -59 F
--------------------------------
E -9 -13 G SOURCE (0[3])
-9[24] -33 -49 A
--------------------------------
F -11 -1* A F<-B,D,F,H,J,L
-11[24] -35 -13* G
-59 -11* F
-83 -31* D
-107 -5* C
-179 -67* J
-251 -47* L
-347 -65* I
--------------------------------
G -13 -19 J G<-A,B,E,F,I,J
-13[24] -37 -55 -7 D
--------------------------------
H -15 -11 F SOURCE (0[3])
-15[24] -39 -29 C
-63 -47 L
-119 -89 -17 I
--------------------------------
I -17 -25 A I<-B,D,F,H,J,L
-17[24] -41 -61 G
--------------------------------
J -19 -7* D J<-B,C,F,G,J,K
-19[24] -43 -1* A
-91 -17* I
-115 -43* J
-139 -13* G
-187 -35* F
-283 -97* -5 C
--------------------------------
K -21 -31 D SOURCE (0[3])
-21[24] -45 -67 -19 J
--------------------------------
L -23 -17 I L<B,D,F,H,J,L
-23[24] -47 -35 F
-71 -53 -5 C
-95 -71 L
The above table and the derived state transition diagram are full of faults. The following response from Mensanator gives an excellent set of corrections.
This has numerous ommissions and errors:
State Transition Status
A A Ok
G Ok
E ERROR (arrowhead wrong, s/b E --> A)
B A Ok
C Ok
D MISSING
F MISSING
G MISSING
I MISSING
J Ok
L MISSING
C D Ok
J Ok
D C Ok
L Ok
I Ok
F MISSING
G ERROR (no transition from D to G)
E A ERROR (arrowhead wrong, s/b E --> A)
G Ok
F A Ok
C MISSING
D MISSING
F Ok
G Ok
I MISSING
J MISSING
L MISSING
G D Ok
J Ok
H F Ok
C Ok
L Ok
I MISSING
I A Ok
G Ok
K ERROR (no arrowhead, but I -- K is invalid either way)
J A Ok
C MISSING
D Ok
F MISSING
G MISSING
I Ok
J MISSING
L Ok
K D Ok
J Ok
F ERROR (no transition from K to F)
I ERROR (no arrowhead, but I -- K is invalid either way)
L I Ok
F MISSING
C Ok
L MISSING
The diagram produced from the predecessor sets immediately reveals, in addition to the expected A<-A loop an additional loop, F<-F. The directions of the edges in the diagram are indicated by nested O's at the successor end. The "leaf nodes" bear an asterisk in this diagram.
The diagram is very complex (possibly as a result of overkill in the selection of transitions to be included), but a number of possible loops are easily discovered in it whose states are presented in red. One cycle in the negative integers consists of I, A, G, D, I, G, J, I which numerically goes -17, -25, -37, -55, -41, -61, -91, -17. A second cycle consists of D, C, D which numerically goes -7, -5, -7. Interestingly, D occurs in both loops, but each loop goes its separate way depending on whether D is -7 (to C) or -55 (to I). I behaves similarly with separate values of -41 and -17. These anomalies may simply be saying that the state descriptors have not been optimally selected. There are some minor omissions in the diagram, but the points which need to be made can be made with the diagram as it is.
Several potential loops in the state transition diagram seem not to
be realized with numeric examples. For example, the L, I, A, G, D,
L loop, shown in red because of its potential, does not get
get expressed. That raises a possibility -- that even though a
state transition diagram indicates the possibiliy of a loop, that
possibility might not be realized by numeric pursuit of pathways in the
actual itineraries.
This demonstrates that the state transition diagrams for the negative integers differs from that for positive integers in several major features. In particular, that for negative integers allows loops while that for positive integers does not allow loops, thus demonstrating that behaviors observed among negative integers cannot be used to argue that those same features will appear in the positive integers.