**The 3 n+1 Problem**

The* *3*n+*1 problem appears to have been first posed
by Collatz in 1937. In this problem, a sequence is generated starting with
an initial natural number *n*. The rule for generating the next natural number in the sequence is; if
*n* is even, the next natural number in the sequence is *n*/2, or if
*n* is odd, the
next natural number in the sequence is 3*n*+1. For example, if the
initial value of *n* is 17, the sequence generated is {17, 52, 26, 13, 40, 20, 10, 5, 16,
8, 4, 2, 1, 4, 2, 1, ...}. If the natural number 4 is
encountered in the sequence, the sequence starts repeating ({4,
2, 1} is generated over and over). There are three possibilities; (1) the
natural number 4 is encountered in the sequence, (2) the sequence starts
repeating, but for a different cycle than {4, 2, 1}, or (3) the sequence
doesn't repeat (in which case the natural numbers generated in the sequence have
to keep getting larger and larger). The 3*n*+1 conjecture states that only
the first possibility can occur. Let *c* be an odd integer greater than or
equal to -1 that is not divisible by 3. The next number in the 3*n*+*c*
sequence is defined to be 3*n*+*c* if *n* is odd, or *n*/2 if* n
*is even (the next number
in the sequence is always a natural number). The sequence
repeats for {4*c*, 2*c*, *c*} if *c*>-1 or
{2, 1} if *c=-*1. In
this more general sequence, there are usually cycles other than {4*c*,
2*c*,
*c*}. Although the 3*n*+1 and 3*n*-1 sequences have some
unique properties, the 3*n+c* problem is essentially the same for all* c*
values.

The 3*n*+1 problem is difficult and is not likely to be solvable by an
amateur. Although it might seem that the problem should be classified as
"recreational" mathematics, there is a considerable body of mainstream
mathematical literature on the subject. (See Jeffrey C. Lagarias' "The 3*x*+1
problem: An annotated bibliography (1963-1999)" at
http://arxiv.org/abs/math.NT/0309224
and "The 3*x*+1 Problem: An Annotated Bibliography, II (2000-2009)"
at http://arxiv.org/abs/math.NT/0608208.
Also, see Lagarias' 1985 article "The 3*x*+1 problem and its
generalizations" at
http://www.cecm.sfu.ca/organics/papers/lagarias/index.html.) This
article reviews some of the highlights of the 3*n*+1 problem literature
and presents some original research on the matter (in the form of empirically
derived "results"). (For the record, I have a degree in mathematics, but
am not a professional mathematician [as is evident from the lack of formal
proofs].) The level of the presentation is elementary and empirically
derived propositions are specifically identified when practical.
(Sometimes entire sections are mostly empirically derived. In these cases,
a statement to that effect is made at the beginning of the section.)
Interested readers can verify these propositions for themselves, although fairly
sophisticated computer programming skills would be required in a few instances.

Newcomers to the 3*n*+1 problem are frequently mystified by how the
sequence increases and decreases in a seemingly random fashion, but always
appears to arrive at 1. The fluctuations of the sequence are of little
interest once the probability of the situation is taken into account (whether
there are cycles other than {4, 2, 1} is far more interesting). See the
section "A probabilistic heuristic" in the Wikipedia article at
http://en.wikipedia.org/wiki/Collatz_conjecture for a brief introduction to
the subject, the section "A heuristic argument" in Lagarias' 1985 article for
more details, and the section "A Random-Walk Argument" in Richard E. Crandall's^{1}
1978 article "On the "3*x*+1" Problem". The probabilistic argument to be given here avoids the "mixing" assumptions in
other approaches and is amenable to empirical verification. If the 3*n*+1
sequence is bounded and there are no cycles other than {4, 2, 1}, then the
sequence element 1 must eventually be reached. The argument to be given is
then that the 3*n*+1 sequence is bounded.

**A Probabilistic Argument that the 3 n+1 Sequence is Bounded**

The probabilistic argument entails a restructuring of the Collatz graph and
an alternate definition of the 3*n*+1 sequence. The Collatz graph is a tree-like structure that shows how the sequence
element 1 can be arrived at starting from different initial *n* values.
(See the Wikipedia article for the standard depiction of the Collatz graph.
Of course, the graph can only be constructed if the 3*n*+1 conjecture is
assumed to be true.) When
*n* is even and *n*-1 is divisible by 3, there is a node in the graph where the
previous elements in the sequence are 2*n* and (*n*-1)/3. For example, in the Collatz graph, two limbs ending in 32 and 5 converge to form a limb starting at
16. In the restructured Collatz graph, 5 is considered to be a
continuation of the limb segment starting with 16. In general, (*n*-1)/3 is
considered to be a continuation of the limb segment at such nodes. (That
is, "odd" paths are taken when tracing back through the nodes.) The
limbs in the restructured Collatz graph are then;

{..., 24, 12, 6, 3, 10, 5, 16, 8}

{..., 72, 36, 18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20}

{..., 120, 60, 30, 15, 46, 23, 70, 35, 106, 53, 160, 80}

{..., 168, 84, 42, 21, 64, 32}

.

.

.

Each limb other than (4, 2, 1) contains exactly one odd
element divisible by 3. Let *j* denote the odd natural number that is
divisible by 3. The natural numbers to the right of *j* are not divisible by
3 (since 3 does not divide 3*j*+1, (3*j*+1)/2, ...). The natural numbers to
the left of *j* must be of the form 2* ^{i}j* where

The 3*n*+*c* sequence can
be defined by the recurrence operation [(3/2)^{h}(*n*+*c*)-*c*]/2→n, 2^{h
}divides *n*+*c*, 2^{h+1 }does not divide *n*+*c*. Each iteration of the
recurrence operation starting with an odd element will be referred to as a "jump". If *n* is
odd and 2^{2} does not divide *n*+*c*, then the sequence {*n,* 3*n*+*c*,
(3*n*+*c*)/2=(3/2)(*n*+*c*)-*c*, ...} is generated where every other element of the
sequence up to (3/2)(*n*+*c*)-*c* is odd. Similarly, if*
n *is odd and 2^{2}
divides *n*+*c*, 2^{3} does not divide *n*+*c*, then the sequence
*{n,* 3*n*+*c*,
(3*n*+*c*)/2, 3[(3*n*+*c*)/2]+*c*, [3[(3*n*+*c*)/2]+*c*]/2=(3/2)^{2}(*n*+*c*)-*c*, ...} is
generated where every other element of the sequence up to (3/2)^{2}(*n*+*c*)-*c*
is odd. In general, if 2^{h} divides *n*+*c*, 2^{h+1} does not
divide *n*+*c*, then the first element in the sequence that is divisible by 4 is the
one just before (3/2)^{h}(*n*+*c*)-*c*. If [(3/2)^{h}(*n*+*c*)-*c*]/2
is even, then the first element in the sequence that is divisible by 8 has been
found. Each limb of the restructured Collatz graph other than {4, 2, 1}
consists of a series of jumps, the last jump ending in an even natural number
(which connects the limb to the rest of the tree). The question of whether
the 3*n*+1 sequence (or the 3*n*+*c* sequence) is bounded then reduces to the question
of whether there are any jumps ending in an even natural number. The
distribution of the number of jumps (denoted by *i*) before an even natural number
is reached for the 1000000 sequences starting with 3, 9, 15, ..., 5999997 is;

The probability that* i *iterations are required is about (1/2)^{i}. Even though the probability that the sequence is unbounded is effectively 0,
it's unlikely that this part of the 3*n*+1 conjecture is provable.
(The data indicates that the process which forms 3*n+*1 sequences [having
1-2 sequence vectors] consisting of an arbitrarily large number of jumps is
random. [In the mathematical literature, such processes are usually said
to be "pseudo-random", but in the absence of a rigorous definition of "random",
one could argue that saying the process is random is acceptable.]). The
probability that *h=*1 is about 1/2, the probability that* h*=2 is
about 1/4, the probability that* h=*3 is about 1/8, etc. These are
the expected probabilities. Even if such a limb (one having a 1-2 sequence
vector starting with an odd natural number divisible by 3) attaches to another
such limb, and that limb attaches to another such limb, etc., there is no
guarantee that the trunk of (4, 2, 1) would be reached. The 3*n*+1
sequence may then be unbounded in this way. About 90% of the time, the odd
natural number divisible by 3 in a limb that attaches to another limb is larger
than the odd natural number divisible by 3 in that limb. (The proportions
for the first 10, 100, 1000, 10000, 100000, 10000000, and 10000000 odd natural
numbers divisible by 3 are 0.9, 0.91, 0.905, 0.9022, 0.90333, 0.903506, and
0.903254 respectively. The limb (..., 24, 12, 6, 3, 10, 5, 16, 8) doesn't
attach to another limb containing an odd natural number divisible by 3, but this
is still counted as if 3 were greater than an odd natural number divisible by 3
in another limb.) There is then a strong tendency for the limbs to
eventually be attached to the trunk.

If *c*=1 and negative *n* values are allowed, the absolute value of
the integer divisible by 3 in a limb that attaches to another limb is larger
than the absolute value of the odd integer divisible by 3 in that limb about 90%
of the time. Also, the probabilities for the number of jumps required to
reach an even integer starting with an odd integer divisible by 3 are the same as
when only positive* n* values are allowed. If* c*>1
and negative* n* values are allowed, the same probabilities apply.
Let* t* denote the odd integer divisible by 3. For example, for* c*=5,
5 does not divide* t*, -10^{k}≤(*t*-3)/6≤10^{k},
and*
k*=1, 2, 3, ..., 7, the proportions are 0.941176, 0.913043, 0.906933,
0.903944, 0.903457, 0.903540, and 0.903280 respectively.^{
} For* k*=4, a histogram of the differences in absolute values of*
t* values divided by 1536 (to scale the values) is;

For *c*=1 and *k*=4, a histogram of the differences in absolute
values of *t* values divided by 1536 is;

(There are fewer values for* c*=5 due to excluding* t* values
divisible by 5.)* *In general, 8/9 appears to be a lower bound of these proportions.

In 1972, John H. Conway^{2} showed that a more general function
iteration problem similar in form to the 3*n*+1 problem is computationally
undecidable. This lends some credence to the notion that this part of the
3*n*+1 problem is unprovable. However, some progress can be made in
this area; in 1976, Riho Terras^{3} showed that almost all numbers have
finite "stopping time". Let* S*_{0}=*N* and* S _{i}*=

**Necessary and Sufficient Conditions for Cycles in the 3n+c Sequence to Exist
**An element

**The 3 n+c Sequence in the Integer Domain **

In the mathematical literature, the cycles {4, 2, 1} for* c*=1* *
and {2, 1} for* c*=-1 are usually said to be "trivial" (but there doesn't
appear to be any rationale for this designation). For a given* c*
value, there appear to be few (if any) non-trivial cycles.*
*When *c* is composite, some of the cycles that occur are essentially no
different from the cycles that occur for a factor of *c* (the elements of the
cycle are just a common multiple of the elements of the cycle for the factor of
*c*). (Cycles that are not multiples of other cycles are said to be
primitive. Note that the cycle {4*c,* 2*c, c*} for* c*>1
is not primitive.) Also, some cycles for a given *c* value are interrelated; they
have the same lengths (number of elements) and the same number of odd elements (and approximately the same dynamic
range of elements). Counting these types of cycles as redundant, a histogram
of the apparent number of cycles (including the "trivial" cycles) for *c* values less than or equal to 151 is;

0 0

1 20

2 21

3 10

4 1

5 0

6 0

total=52

A Poisson probability distribution with a parameter of 1 can be used to model
the number of 3*n+c* cycles for a given* c* value (see the author's
"The 3*n*+1 Problem: A Probabilistic Approach" at
http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Cox/cox10.pdf).

Associated cycles are defined in the *Journal of Integer Sequences *
article. A Poisson probability distribution with a parameter of 1.082
superimposed on a histogram of the number of cycles* *for a given*
c* value* *(allowing negative* n* values and* *counting only one of
interrelated or associated cycles) for *c* values greater than 0 and less than or equal to
19999 is;

(The mean of the actual distribution is 1.082.)
A Poisson probability distribution with a parameter of 1.070 superimposed on a
histogram of the number of cycles for a given* c* value (allowing negative*
n* values and* *counting only one of interrelated or
associated cycles) for *c* values greater than 0 and less than or equal to
29999 is;

When *c*=-1, there are two known cycles other than (2, 1); the cycle (20, 10, 5, 14,
7) and the cycle (68, 34, 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182,
91, 272, 136). A frequently asked question is why there are non-trivial
cycles for the 3*n*-1 sequence, but apparently not for the 3*n*+1
sequence. There are many such examples when general negative *c*
values are allowed. For example, there is at least one primitive cycle for
the 3*n*+7 sequence, but apparently none for the 3*n*-7 sequence.
However, there appears to be at least one primitive cycle in either the 3*n*+*c*
sequence or the 3*n*-*c* sequence. This indicates that the
distinction between the 3*n*+*c* and 3*n*-*c* sequences is
artificial and that negative* n* values should be allowed (and that the*
c* values should be required to be positive). (However, it is sometimes
convenient to use the original definition of the 3*n*+*c* sequence.)
More evidence that the 3*n*+1 and 3*n*-1 sequences (*n*>0) should
be considered to be the same sequence will be presented later.

**Interrelated 3 n+c Cycles**

In this section, negative *n* values are allowed, *c* is required to
be positive, and the element after an odd element *i *in the 3*n*+*c*
sequence is defined to be (3*i*+*c*)/2. A parity vector gives
the order of even and odd elements in a 3*n+c* sequence, a "'1" for an odd
element and a "0" for an even element. Let* k*_{0},* k*_{1},*
k*_{2}, ...,* k _{m}*

(1) A 3*n+c* cycle exists only if *c* divides 2^{l}-3^{m}.

If this proposition holds, *Z* need not be considered when searching for
cycles; only the factors of 2^{l}-3^{m} need be
considered. When *c*=|2^{l}-3^{m}|, an
arbitrarily constructed parity vector with a length of *l* and containing
*m* 1's corresponds to a cycle (or possibly duplicated sub-cycles if *l*
and *m* are not relatively prime), but the cycle is not guaranteed to be
primitive. When reduced, such a cycle corresponds to a cycle for a *c*
value that is a proper divisor of 2^{l}-3^{m}.
In this sense, there is no problem of finding cycles; they're all well-defined
and determined by the parity vectors. Even the number of interrelated
cycles is determined by the combinatorics of generating parity vectors that are
distinct under rotation (see the above* Journal of Integer Sequences *
article for a table of the numbers of distinct parity vectors for* l* less
than or equal to 20 and* m* less than or equal to 10). There is,
however, the problem of determining which *c* values the primitive cycles
map to. For example, for *l*=1 and *m*=1, the
parity vector is (1) (corresponding to the cycle {-1} for *c*=1), for *l*=2
and *m*=1, the parity vector is (0, 1) (corresponding to the cycle {2, 1}
for *c*=1), and for *l*=3 and *m*=2 the parity vector is (1, 1,
0) (corresponding to the cycle {-5, -7, -10} for *c*=1). For *l*=11
and *m*=7, there are 30 distinct parity vectors, the first few of which are
(0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0), (1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0), (1, 1, 0,
1, 0, 1, 1, 1, 1, 0, 0), ... (corresponding to cycles for *c*=139).
The cycle corresponding to the first parity vector is not primitive (when *c*=139)
and corresponds to the remaining known *c*=1 cycle of {-34, -17, -25, -37,
-55, -82, -41, -61, -91, -136, -68}.
Another example factorization will be given. For *l*=8 and *m*=4,
the distinct parity vectors are (1, 1, 0, 0, 1, 1, 0, 0), (0, 1, 0, 1, 0, 1, 0,
1), (1, 0, 1, 1, 0, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0,
0), (0, 1, 1, 0, 1, 1, 0, 0), (1, 0, 0, 1, 1, 1, 0, 0), (0, 1, 0, 1, 1, 1, 0,
0), (1, 0, 1, 0, 1, 1, 0, 0), (1, 1, 0, 1, 0, 1, 0, 0), and (0, 1, 1, 1, 0, 1,
0, 0) (corresponding to cycles for *c*=175). The first parity
vector corresponds to two duplicated primitive cycles for* c*=7, the second
parity vector corresponds to four duplicated primitive cycles for* c*=1, the third parity vector
corresponds to a primitive cycle for *c*=25, the fourth and fifth parity vectors
correspond to primitive cycles for* c*=35, and the remaining parity vectors
correspond to primitive cycles for* c*=175. Although 5 also
divides 175, there are no primitive cycles for this* c* value when* l*=8
and* m*=4 (the parity vectors have been used up by cycles for the larger *c*
values and* c*=1).

Apparently, every* c* value is covered in this mapping. A more
appropriate question to ask than why cycles don't exist for a given* c*
value is why they do exist and what the expected number of cycles is (see the*
Journal of Integer Sequences *article for more details).

(For a given *c* value, two primitive cycles cannot have the same parity
vector. For example, the 3*n*+5 sequence 83, 254, 127, 386, 193, 584,
292, 146, and 73 has the same parity vector as the 3*n*+5 cycle 19, 62, 31,
98, 49, 152, 76, 38, and 19. The ratios of the corresponding odd elements
decrease monotonically and are 4.368, 4.097, 3.939, and 3.842. More
generally, a primitive 3*n*+*c*_{1} cycle cannot have the same
parity vector as a primitive 3*n*+*c*_{2} cycle. For
example, the 3*n*+7 sequence 65, 202, 101, 310, 155, 472, 236, 118, and 59
has the same parity vector as the above 3*n*+5 cycle, but the ratios of the
corresponding odd elements are 3.421, 3.258, 3.163, and 3.105. The 3*n*+7
sequence 1, 10, 5, 22, 11, 40, 20, 10, and 5 has the same parity vector as the
above 3*n*+5 cycle, but the ratios of the corresponding odd elements are
0.053, 0.161, 0.224, and 0.263. For a primitive
3*n*+*c* cycle where *c*≠|2^{l}-3^{m}|,
there is always another cycle [not necessarily primitive] having the same parity
vector where *c*=|2^{l}-3^{m}|. This
accounts for Proposition (1).)

**The Minimum Element in a 3 n+c Cycle **

In 1997, Halbeisen and Hungerbühler^{6 }
proved optimal estimates for the length of a Collatz cycle in terms of its
minimum using the formula *M _{l,m}*=∑(]

Let* s* = (s_{1},...,* s _{l}*)
and

* S _{l}*

Let *n ≤ l* be natural
numbers. Let* s*̃_{i }= ]*in*/*l*[ - ](*i *-
1)/*l*[ (for 1≤ *i* ≤ *l*). Then *φ*(*s*̃) equals
the minimum* φ*(*t*) value of * s*̃ (equal to *M _{l,n}*).

(Due to typographical difficulties, the exact form of this lemma is not
duplicated here.) In the proof of this lemma, Lemma 4 is used and ∑^{k}_{i}_{=1}*t _{i}* is represented by a staircase. For example, a
staircase for

The staircase using the floor function (not used in the proof of Lemma 5) can be viewed as being an upside-down
staircase where Halbeisen and Hungerbühler's logic
can be used to find a lower bound of the maximum odd element in a Collatz cycle.
As will be proved, twice the minimum element in such a cycle (one having the
parity vector ]*jm/l*[ - ](*j*-1)*m/l*[, *j*=1, 2, 3, ...,
*l*) must be larger than the maximum odd element. Let *t _{j}=*[

(2) If *c*=|2^{l}-3^{m}| and the elements of the interrelated 3*n*+*c* cycles are
positive, *M _{l}*

For example,
when* l=*11, and* m*=7,* M _{l}*

The largest upper bound of minima in potential 3*n+c* cycles having the same number of even
elements can also be easily determined. An empirical result is;

(3)* cM _{l}*

For example,* M*_{6,4}/|2^{6}-3^{4}| is greater
than *M*_{5,3}/|2^{5}-3^{3}|, 2^{5}-3^{3}
is positive, and 2^{6}-3^{4} is negative (the maximum occurs at
the sign change). *M*_{8,5}/|2^{8}-3^{5}| is
greater than *M*_{9,6}/|2^{9}-3^{6}|, 2^{8}-3^{5}
is positive, and 2^{9}-3^{6} is negative (the maximum occurs
before the sign change).

**The Minimum Element in a 3 n+c Cycle and the Continued-Fraction Convergents of
Log(3)/Log(2)**

In his 1978 article, Crandall proved a lower bound of 17985 for the number of
odd elements in a 3*n*+1 cycle (*n*>0) using the minimum in a
hypothetical cycle and the continued-fraction convergents of log(3)/log(2).
Recent 3*n*+1 cycle research frequently involves use of the continued-fraction convergents of log(3)/log(2). For some purposes, this is too restrictive;
valuable information is thrown away. This
is the rationale for defining generalized continued-fraction convergents
(allowing less accurate approximations). Let (*c*, *d*), (*e*,
*f*), and (*g*, *h*) denote successive continued-fraction
convergents of a real number and let *j* denote the ceiling of *h*/*f*.
(*ei*,* fi*) and* *(*ei*+*c*, *fi*+*d*), *i*=1,
2, 3, ..., *j*, can be considered to be generalized continued-fraction
convergents of the real number. (Sometimes it is convenient to consider
just (*ei*+*c*,* fi*+*d*),* i*=1, 2, 3, ...,* j*,
to be the generalized continued-fraction convergents of the real number;* ei*+*c*
and *fi+d* are then relatively prime.) The generalized continued-fraction convergents of log(3)/log(2) are
(2, 1), (3, 2), (4, 2), (5, 3), (6, 4), (8, 5),
(9, 6), (11, 7), (16, 10), (19, 12), (24, 15), (27, 17), (38, 24), (46, 29),
(57, 36), (65, 41), (76, 48), (84, 53), .... An empirical result is;

(4) The absolute values of 2^{l}-3^{m}
increase monotonically for (*l*, *m*) values that are generalized
continued-fraction convergents of log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4),
and (9, 6)).

The significance of this is that for a given *c* value, there can be at
most one solution of *c*=|2^{l}-3^{m}| where
(*l*, *m*) are generalized continued-fraction convergents of
log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4), and (9, 6)).

An algorithm for generating the lengths and numbers of odd elements of 3*n*+1
or 3*n*-1 (*n*>0) sequences associated with potential cycles having
1-2 sequence vectors almost always gives the generalized continued-fraction
convergents of log(3)/log(2) (and only these values). The algorithm gives
four (*l*,* m*) values that are not generalized continued-fraction
convergents of log(3)/log(2) and four generalized continued-fraction convergents
of log(3)/log(2) are not given. * *(This algorithm will be
discussed in detail later. See the section "The Minimum in a Collatz
Cycle".) This is more evidence that the generalized
continued-fraction convergents of log(3)/log(2) are of significance to the 3*n*+1
problem. The continued-fraction convergents of log(3)/log(2) will be
further generalized in the next section.

*M*-Cycles and an Algorithm for Determining When the Sign of 2^{l}-3^{m}
for a Fixed* l-m* Value Changes

A cycle with no even elements can occur only if (3/2)^{i}(*n*+*c*)*-c*=*n*
where *n* is odd, *i*<*h*, and 2* ^{h }*is the largest power of 2 that
divides

(5) A cycle in the 3*n*+*c* sequence having only one even
element can occur only if* c*=|2^{l}-3^{l-1}|.

(6) A cycle in the 3*n*+*c* sequence having only one odd element
can occur only if* c*=|2^{l}-3|.

A successor of *a*, *a*>0, in the 3*n*+*c* sequence is
greater than *a* if 3* ^{m}a*>2

(7) These (*l*,* m*) values include the generalized continued-fraction
convergents of log(3)/log(2) (except for (2, 1), (4, 2), (6, 4), and (9, 6)).
Also, 2*m*>*l*.

(8) The
difference in *l* values of a pair of successive *M*-cycles is 2 or 3
and the difference in *m* values is 1 or 2 respectively.

(9) There are either 3 or 4 successive *M*-cycles where the
difference in *l* values is 3 (2^{l}-3^{m} is
negative for the third or fourth *M*-cycle and positive otherwise).

(10) When there are 4
successive *M*-cycles where the difference in *l* values is 3, the
values of* M _{l,m}*/|2

(11) There are no successive *M*-cycles where the difference in *l*
values is 2.

A table of the ]*jm*/*l*[ -](*j*-1)*m*/*l*[, *j*=1,
2, 3, ..., *l*, values (given in the columns) for (*l*, *m*)=(3,
2), (5, 3), (8, 5), ..., (103, 65) is;

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 0

Note that there are rows of values consisting of all 0's, rows of values
consisting of all 1's, rows of values consisting of all 1's except for an
initial value of 0, and pairs of adjacent rows of values where the element-by-element sum
consists of all 1's (disregarding the first element of the first row). The elements in
one of these pairs of adjacent rows appear to become all 0's or 1's eventually.
Let *q _{j}*

(12) A rotation of *r*
matches the first *l*_{2} elements of the parity vector *p*
(usually, a right-rotation of 3 suffices).

The absolute values of 2^{l}-3^{m} where (*l*,
*m*) are multiples of a continued-fraction convergent (as usually defined)
of log(3)/log(2) increase monotonically since the values of* M _{l}*

l

l

.

.

.

The initial difference between successive (

Empirical results (showing why the parity vector

(13) If the maximum

(14) If the maximum

Let

(15) The

For example, (149, 94) is a generalized continued-fraction convergent of log(3)/log(2) where

Upper bounds of minima in 3

**Miscellaneous Properties of Minima in 3 n+c Cycles**

When the element following an odd element *i* in the 3*n*+*c*
sequence is defined to be (3*i*+*c*)/2 instead of 3*i*+*c*,
1-2 sequence vectors become 0-1 sequence vectors. Note that 2*m* must
be greater than* l* for there to be any cycles having 0-1 sequence vectors.
For example, when *l*=11 and *m*=7, the parity vectors of cycles
having 0-1 sequence vectors are (1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1), (1, 1, 0, 1,
1, 0, 1, 0, 1, 1, 0), (1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1), (1, 0, 1, 1, 0, 1, 0, 1,
0, 1, 1), and (1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1) (corresponding to primitive
cycles for *c*=139). The 3*n*+139 cycle corresponding to the
second parity vector is {-4151, -6157, -9166, -4583, -6805, -10138, -5069,
-7534, -3767, -5581, -8302} and twice the largest element in the cycle (-3767)
is greater than the smallest odd element (-6805). In the other
interrelated 3*n*+139 cycles having 0-1 sequence vectors, twice the largest element in a cycle is not greater than the
smallest odd element in the cycle. Let* min* denote the element of a
cycle in the 3*n*+*c* sequence having the smallest absolute value and
let *max* denote the odd element having the largest absolute value.
More
empirical results are;

(16) If* c*=|2^{l}-3^{m}| and 2*m*>*l*,
there is at least one primitive cycle having a 0-1 sequence vector and there are
usually primitive cycles not having 0-1 sequence vectors. * *If*
c*=|2^{l}-3^{m}|, 2*m*>*l*,
and
the greatest common divisor of *l* and *m* is 1, there is at least one
cycle among the interrelated primitive cycles having 0-1 sequence vectors where 2|*min*|>|*max*|
(there is no cycle among the interrelated primitive cycles not having 0-1 sequence
vectors where 2|*min*|>|*max*|).
If *c=*|2^{l}-3^{m}|, 2*m*<*l*, and
*gcd*(*l*, *m*)=1, there is at least one cycle among the
interrelated primitive cycles where 2|*min*|>|*max*|.

(17) 2|*min*|>|*max*| for a primitive 3*n*+*c* cycle where* gcd*(*l*,* m*)=1*
*only if *c*=|2^{l}-3^{m}|.

Usually, if *c*=|2^{l}-3^{m}|, 2*m*>*l*,
and* gcd*(*l*,* m*)=1,* *there is exactly one cycle among
the interrelated primitive cycles having 0-1 sequence vectors where 2|*min*|>|*max*|.
When *c*=1675, *l*=9, and *m*=7, there are two primitive cycles
where 2|*min*|>|*max*|. For one of these cycles,* min=-*2219,*
max*=-4429, and 2|*min*| is barely greater than |*max*| (*min*=-2363
and* max*=-3997 for the other* *cycle).* *Usually, if *c*=|2^{l}-3^{m}|,
2*m*>*l*, and *gcd*(*l*, *m*)≠1,
there is at least one cycle among the interrelated primitive cycles having 0-1
sequence vectors where 2|*min*|>|*max*|. When *c*=9823, *
l*=14, and *m*=8, there doesn't appear to be a primitive cycle where 2|*min*|>|*max*|.
If* c*=|2^{l}-3^{m}|, 2*m*<*l*, and*
gcd*(*l*,* m*)=1, there appears to be exactly one cycle among the
interrelated primitive cycles where 2|*min*|>|*max*|. If* c*=|2^{l}-3^{m}|,
2*m*<*l*, and* gcd*(*l*,* m*)≠1, there doesn't appear
to be a primitive cycle where 2|*min*|>|*max*|. Usually, if a
primitive 3*n*+*c* cycle where* c* properly divides 2^{l}-3^{m}
exists and* gcd*(*l*,* m*)≠1, then 2|*min*| is not greater
than |*max*|. When* c*=145,* l*=12, and* m*=8,* min*=-617,*
max*=-1231, and 2|*min*| is barely greater than |*max*|.
Also, when* c*=493,* l*=12, and* m*=8,* min=-*1829,* max*=-3635,
and 2|*min*| is barely greater than |*max|*.

A graph of the proportions of 3*n+c* cycles having 0-1 sequence vectors for
*c* less than or equal to 499, 997, 1499, 1999, ..., 9997 is;

**More on m-Cycles**

(18) There exists at least one primitive 3

Let

(19) The maximum |

For exactly one each of the cycles with

Belaga

(20) The minimum |

(21) |1-

For the 1556 cycles for

(22)

An exception occurs when

(23) If the (

This distribution has a mean, standard deviation, and sample size of -1.3532, 2.4842, and 9332 respectively. A histogram of the

a=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 i=1 3423 253 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1845 1858 182 29 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1003 4035 1254 254 45 8 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 428 3412 2103 791 213 59 28 4 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 267 2542 3113 1688 689 256 101 29 13 6 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 145 1829 3082 2322 1389 707 332 153 74 44 22 10 5 2 1 0 0 0 0 0 0 0 0 0 7 103 1245 2546 2579 1919 1242 726 452 294 161 86 70 37 13 10 5 6 4 1 0 0 0 0 0 8 66 886 2149 2488 2066 1539 1183 851 592 363 306 166 125 76 47 39 24 14 13 8 7 3 4 2 9 45 602 1590 2012 1968 1770 1461 1089 913 703 516 410 252 212 161 132 99 80 49 35 34 18 12 13 10 25 397 1161 1592 1662 1625 1526 1335 1051 914 775 590 482 389 332 257 210 173 146 118 81 88 80 61 11 31 297 726 1116 1340 1345 1310 1250 1206 1048 893 735 641 543 485 393 331 311 267 217 215 156 134 121 12 13 193 483 782 943 1081 1140 1117 1041 923 932 755 704 634 551 518 494 423 411 283 308 294 257 220 13 11 139 326 505 661 750 869 856 866 815 802 727 696 609 603 534 552 521 442 455 379 381 321 287 14 8 96 193 324 448 596 616 700 697 696 664 626 609 602 552 547 523 494 501 397 402 423 355 363 15 6 49 135 222 318 368 462 501 506 503 519 474 473 491 489 524 485 479 413 412 435 409 334 369 16 7 43 78 127 197 261 320 343 363 372 419 397 410 411 388 417 417 386 400 362 370 355 349 305 17 4 18 67 95 123 151 205 229 232 283 260 264 276 322 309 313 318 316 312 290 315 309 298 255 18 3 10 34 58 66 117 136 164 178 185 200 216 214 217 216 227 264 257 260 261 229 234 234 259 19 1 6 17 36 59 62 82 90 124 141 131 134 153 138 155 158 169 172 178 208 193 185 185 167 20 2 8 15 23 34 54 54 92 65 64 88 93 103 115 129 134 138 138 152 160 127 146 131 129 21 1 1 10 22 26 29 38 43 63 49 65 98 76 74 106 96 92 110 111 114 101 110 105 115 22 2 4 5 5 10 26 25 34 42 44 44 58 55 65 64 62 70 65 74 72 75 68 75 70 23 0 2 1 7 14 6 17 19 18 21 33 32 36 29 40 42 41 53 48 56 57 55 51 65 24 0 0 1 5 4 6 8 9 12 18 16 24 21 28 21 30 25 40 26 42 36 38 22 37 25 1 0 0 2 2 3 5 9 8 13 13 17 15 19 20 21 28 25 24 30 20 27 20 25 26 0 0 0 1 1 4 5 2 4 8 11 9 7 11 8 15 12 11 17 21 17 21 22 18 27 0 0 3 0 2 2 4 5 7 6 8 3 9 10 4 9 15 11 18 10 14 15 8 7 28 0 0 0 0 1 0 2 4 4 3 0 3 7 4 5 9 9 7 8 7 4 6 9 16 29 0 0 0 1 2 2 0 1 0 1 1 4 2 5 4 5 5 7 8 6 5 7 2 8 30 0 0 0 0 0 0 1 1 1 2 1 3 1 2 3 3 2 2 3 2 1 3 2 5 31 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 3 4 2 0 0 0 3 0 32 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 1 0 0 2 3 1 1 0 3 33 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 2 0 0 2 3 34 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 2 37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 38 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0For

A linear least-squares fit of the means plotted against log(

For

An empirical result (based on the 663743 cycles with attachment points for

a=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 i=1 6226 4689 984 224 43 11 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1214 10462 9699 5654 2684 1338 697 332 154 92 49 21 10 12 1 2 1 0 0 0 0 0 0 0 3 0 2774 7635 9026 8045 6440 5074 3883 2929 2135 1716 1270 913 701 571 466 366 316 224 169 142 115 101 73 4 0 0 961 2121 3161 3788 4081 4052 3908 3562 3303 2779 2494 2237 2015 1784 1730 1471 1353 1132 1067 1001 822 782 5 0 0 0 61 277 487 774 1043 1249 1428 1495 1530 1583 1572 1555 1555 1487 1517 1457 1373 1265 1278 1161 1135 6 0 0 0 0 0 5 30 73 134 169 235 298 383 462 499 607 644 664 704 727 742 762 710 686 7 0 0 0 0 0 0 0 0 1 2 7 24 29 38 62 79 102 129 139 157 198 187 205 227 8 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 8 10 14 14 10 16 21 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2The means and standard deviations of these distributions for

0 < Λ < ∑1/

(26) 3

This proposition sometimes fails for cycles with one or two primary attachment points and having (

Simons and de Weger's Corollary 5 (of Lemma 4) is;

0 < Λ <

Λ >

Using Corollary 5 gives Simons and de Weger's Corollary 13 (where the global minimum of an

Based on this plot,

A histogram of the [3·log(

This distribution has a mean of 29.2260 and a standard deviation of 4.3270. The only negative value (-1) occurs for the

(27) [3·log(

This proposition is true for 99.9235% of the cycles. (Current

For larger

A plot of the means, standard deviations, and expected means of the distributions of differences versus log(

(The curves for the means and expected means intersect.) Although the mean starts out being smaller than the expected mean and eventually becomes larger than the expected mean, the curve of means still appears to be mostly linear (when plotted against log(

The non-linearity of the standard deviation curve is due to a large number of cycles with attachment points (2849) for

p

For these

A quadratic least-squares fit of the number of cycles plotted against (

(The coefficients are poorly conditioned when the number of cycles is plotted against

A linear least-squares fit of the proportions of cycles in the left-hand tail of the distribution of differences (up to and including cycles for which the difference is 0) plotted against (

Modifying the scaling factor (1/

Even this small of a change in the scaling factor (from 1/1.584962501 to 1/1.5) skews the distribution significantly (there are fewer negative differences and more large positive differences). When

The mean of the distribution of differences still appears to increase linearly (mostly) with log(

A linear least-squares fit of the means plotted against log(

A plot of [3·log(

A linear least-square fit of the means plotted against log(

The slight deviation in the linearity of the means when plotted against log(

A plot of [3·log(

For this scaling factor, the distribution is skewed to the left. The "best" scaling factor is then somewhere between 1/1.75 and 1/1.5. The best least-squares fit of the means plotted against log(

The [3·log(

For larger

For a fixed

a

i

5 0 0 0 0 0 2 16 19 46

a

i

5 0 0 0 0 0 0 14 19 38

A least-squares fit of the means plotted against log(

An advantage to working with the average of the |

When

(31)

Of the 23132 cycles for

The sample sizes for

A linear least-squares fit of the curve for

p

p

Although a chi-square probability distribution doesn't model the distribution very well for larger

A plot of the distribution of the minimum power of

When the curve for the actual distribution rises more steeply than the curve for the chi-square distribution (when

A plot of the distribution of the minimum power of

Given the number of cycles for an

The Minimum |

# cycles 2 3 4 5 6 7 8

2 n/a 26.498 14.273 8.897 7.540 5.456 3.876

3 n/a n/a 25.309 14.022 10.020 6.825 5.020

4 n/a n/a n/a 24.508 14.577 9.484 7.686

5 n/a n/a n/a n/a 23.872 14.445 9.315

6 n/a n/a n/a n/a n/a 24.387 13.634

7 n/a n/a n/a n/a n/a n/a 18.637

When none of the (

# cycles 2 3 4 5 6 7 8

2 n/a 34.344 18.520 11.294 8.939 6.320 5.110

3 n/a n/a 31.834 16.662 12.025 8.160 6.014

4 n/a n/a n/a 30.582 17.415 11.584 7.892

5 n/a n/a n/a n/a 29.094 16.554 11.606

6 n/a n/a n/a n/a n/a 32.073 18.093

7 n/a n/a n/a n/a n/a n/a 32.363

When none of the (

# cycles 2 3 4 5 6 7 8

2 n/a 41.609 21.938 13.610 10.057 7.595 5.576

3 n/a n/a 38.775 20.521 13.873 9.730 6.622

4 n/a n/a n/a 35.499 20.758 13.006 8.901

5 n/a n/a n/a n/a 35.544 18.489 12.346

6 n/a n/a n/a n/a n/a 34.944 18.806

7 n/a n/a n/a n/a n/a n/a 34.597

When none of the (

p

(

Another way to model the number of cycles is to count only one of interrelated cycles, only one of associated cycles, and only one of cycles where the (

The Number of Prime Factors of 2

A plot of the means of these distributions is;

More on the Domain of 3

When the domain of the absolute values of the

An example of a "jumped-over" attachment point (for

(32) The destination of a jumped-over attachment point is another attachment point if the destination is even.

(33) The sign in an extended sequence can change only once.

If the sign in the extended sequence of a jumped-over attachment point changes, it usually does so at the attachment point. For example, -63→74 for a cycle for

An example of a "no-jump" attachment point (for

(34) In a cycle having an attachment point, there is at least one no-jump or one-jump attachment point.

(36) Except when the last attachment point in primary, secondary, tertiary, etc., attachment points is a jumped-over attachment point, the destination of a jumped-over attachment point is the next attachment point.

(37) A jumped-over attachment point cannot be a primary attachment point.

In the chain equation, the element after an odd element

Solving for a one-jump attachment point followed by a no-jump attachment point (in primary, secondary, tertiary, etc., attachment points) gives the Diophantine equation (3/2)

Hardy and Littlewood

The exponents of 3/2 in the above equations are small. For example, for

(38) The last attachment point in the group of primary, secondary, tertiary, etc., attachment points preceding a primary multiple-jump attachment point is a no-jump or jumped-over attachment point.

A Diophantine equation (similar in form to those derived for primary, secondary, tertiary, etc., attachment points) involving the

Another empirical result is;

(39) A primary attachment point is a multiple-jump attachment point only if the attachment point is one jump away from the first odd cycle element after the preceding primary attachment point.

A consequence of the above proposition (and Proposition (37)) is that a

Let 2

(40) For a one-jump attachment point that is a primary attachment point, the expected value of

(41) For a multiple-jump attachment point that is a primary attachment point,

There is usually a one-jump attachment point in a cycle where

For all i

The expected number of odd elements in the jump (or jumps in the case of a multiple-jump attachment point) from

The mean of the distribution is -0.004901 and the standard deviation is 2.3789. Other than being discrete-valued, the distribution resembles a normal probability distribution for small

A histogram of the number of odd elements in the jump (or jumps) from

This distribution has a mean of -0.1381 and a standard deviation of 2.3462. A plot of the standard deviations of the distributions versus

The distributions appear to be tending towards a fixed probability distribution.

A histogram of the sum of the number of odd elements in the jumps from

The above distribution has a mean of -0.4591 and a standard deviation of 6.6471. A quadratic least-squares fit of the standard deviations of these distributions plotted against log(

The number of jumps from

Let

(42) The

Let

cLet_{max}= 49999 99997 149999 199999 i=0 9.087466e-001 9.103450e-001 9.108970e-001 9.111167e-001 1 6.377740e-002 6.288359e-002 6.253354e-002 6.238482e-002 2 1.576506e-002 1.560997e-002 1.560444e-002 1.560868e-002 3 4.056629e-003 3.968739e-003 3.938980e-003 3.916318e-003 4 9.995667e-004 9.866914e-004 9.902790e-004 9.858214e-004 5 2.581761e-004 2.566830e-004 2.553726e-004 2.519073e-004 6 6.479131e-005 6.738539e-005 6.547580e-005 6.579589e-005 7 1.088098e-005 1.285856e-005 1.365493e-005 1.372299e-005 8 0.000000e+000 9.765999e-007 2.544396e-006 2.819064e-006 9 4.945902e-007 4.882999e-007 5.936925e-007 6.382787e-007 10 0.000000e+000 0.000000e+000 8.481321e-008 5.318989e-008 - 6.320368e-003 5.867575e-003 5.698006e-003 5.652702e-003 n= 2021876 6143765 11790616 18800566

A histogram of the differences between the number of odd elements in primary one-jump attachments points and the

As expected, the shoulders on the peak for the

(44) When

When

(45) When

When

For primary multiple-jump attachment points,

cFor a primary no-jump attachment point, the probability that_{max}= 49999 99997 149999 199999 j=1 5.018008e-001 5.008240e-001 5.004788e-001 5.004156e-001 2 4.099361e-001 4.108239e-001 4.110778e-001 4.110728e-001 3 7.693787e-002 7.689667e-002 7.703393e-002 7.706442e-002 4 9.129327e-003 9.315844e-003 9.303581e-003 9.330295e-003 5 1.855497e-003 1.810723e-003 1.794086e-003 1.802707e-003 6 2.786773e-004 2.807546e-004 2.665826e-004 2.699150e-004 7 5.467719e-005 4.341567e-005 3.855657e-005 3.664347e-005 8 7.055122e-006 3.473253e-006 5.723242e-006 6.233167e-006 9 0.000000e+000 1.157751e-006 9.036697e-007 1.133303e-006 10 0.000000e+000 0.000000e+000 0.000000e+000 1.888838e-007 11 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 n= 566964 1727487 3319797 5294259

A histogram of

A histogram of the sum of the

This distribution has a mean of -5.2770 and a standard deviation of 8.9742. The difference between the sum of the

(The coefficients are poorly conditioned when the means are plotted against

A histogram of

This distribution has a mean of 0.03171 and a standard deviation of 2.0258 (there are 9462 samples). A histogram of the sum of the

This distribution has a mean of 0.07266 and a standard deviation of 3.0704 (there are 4129 samples).

A histogram of

This distribution has a mean of -0.6945 and a standard deviation of 1.9292 (there are 2236 samples). A histogram of the sum of the

This distribution has a mean of -4.0262 and a standard deviation of 4.8975 (there are 764 samples).

A histogram of the sum of the

This distribution has a mean of 12.1873 and a standard deviation of 10.0945. [3·log(

This distribution has a mean of 15.2292 and a standard deviation of 19.0883. [3·log(

1 .0071 .0084 .0102 .0113 .0102 .0101 .0096 .0082 .0083 .0083

2 .0109 .0115 .0127 .0127 .0120 .0113 .0106 .0085 .0083 .0081

3 .0084 .0100 .0098 .0102 .0113 .0118 .0108 .0089 .0089 .0091

4 .0161 .0153 .0159 .0143 .0133 .0131 .0122 .0101 .0098 .0095

5 .0141 .0140 .0153 .0146 .0147 .0144 .0135 .0112 .0114 .0114

6 .0206 .0174 .0169 .0169 .0158 .0151 .0140 .0117 .0117 .0117

7 .0308 .0240 .0216 .0184 .0168 .0168 .0151 .0127 .0122 .0124

8 .0219 .0215 .0196 .0177 .0186 .0179 .0166 .0133 .0129 .0131

9 .0379 .0321 .0296 .0265 .0239 .0232 .0204 .0166 .0165 .0162

10 .0463 .0352 .0333 .0290 .0259 .0243 .0219 .0177 .0171 .0171

11 .0668 .0464 .0392 .0335 .0312 .0282 .0258 .0209 .0198 .0188

12 .0521 .0445 .0392 .0327 .0304 .0279 .0251 .0201 .0197 .0195

13 .0758 .0542 .0506 .0437 .0396 .0373 .0329 .0262 .0248 .0243

As expected, the proportions for the larger [3·log(

This composite curve indicates that the

19 0 0 0 1 1 0

20 0 0 1 2 0 0

21 0 2 5 0 0 0

22 0 12 5 2 0 0

23 11 28 1 0 0 0

24 191 16 0 0 0 0

25 236 1 0 0 0 0

As discussed in the

For

There are also frequently associated cycles (defined and discussed in the

(

[3·log(

2 0 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 1 0

9 0 0 0 0 0 0 0 1 0 0

10 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 1 2 0 0 0

12 0 0 0 0 0 0 0 1 0 0

13 0 0 0 0 1 1 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0

15 0 0 0 1 1 0 0 0 0 0

16 0 0 1 1 0 0 0 0 0 0

17 0 0 0 1 0 0 0 0 0 0

18 0 1 0 1 0 0 0 0 0 0

19 0 3 0 0 0 0 0 0 0 0

20 7 3 1 0 0 0 0 0 0 0

21 14 0 0 0 0 0 0 0 0 0

The associated cycles have the same function as the multiples of cycles; the modal classes of the distributions of [3·log(

For

(

[3·log(

8 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 2

11 0 0 0 0 0 0 1 0

12 0 0 0 0 0 0 0 0

13 0 0 0 0 1 1 1 0

14 0 0 0 0 0 1 0 0

15 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0

17 0 0 0 0 0 0 0 0

18 0 0 1 1 0 0 0 0

19 0 0 1 0 0 0 0 0

20 0 0 0 0 0 0 0 0

21 18 1 0 0 0 0 0 0

22 10 0 0 0 0 0 0 0

The modal classes are 21, 21, 18.5, 18, 13, 13.5, 12 and 10.

For

(

[3·log(

-10 0 0 0 0 0

-9 0 0 0 0 0

-8 0 0 0 0 0

-7 0 0 0 0 0

-6 0 0 0 1 0

-5 0 0 1 0 0

-4 0 0 0 0 0

-3 0 0 0 0 0

-2 0 0 0 0 0

-1 0 0 1 0 0

0 0 0 0 0 0

1 0 0 0 0 0

2 0 0 0 0 0

3 0 0 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

6 0 0 0 0 0

7 0 0 0 0 0

8 0 0 0 0 0

9 0 0 0 0 0

10 0 0 0 0 0

11 0 0 0 0 0

12 0 0 0 0 0

13 0 0 0 0 0

14 0 0 0 0 0

15 0 0 0 0 0

16 0 0 0 0 0

17 0 0 0 0 0

18 0 0 0 0 0

19 0 0 0 0 0

20 0 0 0 0 0

21 0 1 0 0 0

22 3 0 0 0 0

23 7 0 0 0 0

In this case, there are two separate populations of (

(47) For a given

For the 3

(48) For a given

The order of a cycle is defined to be the smallest natural number of the form 3∙2

(49) For a one-jump attachment point, the cycle order equals the extension order.

Let

Let

As

A linear least-squares fit of the means of these distributions plotted against log((

For

Let

(50) If

(51) If

If the primary attachment point is

In this paragraph, cycles containing only one attachment point are discussed. Also, the order of the extended sequence is required to have been determined by 4|

Consider a 3

(52) For a given number of odd elements in a generalized dead limb, the number of even elements is mostly fixed; the number of even elements can deviate by at most 1

(53) The adjusted (

The (

(54) The (

Let

(55) The number of even elements in a generalized dead limb containing

Weibull probability distributions are used to model birth and decay processes and have a scaling and shaping parameter. For the generalized dead limbs where the odd elements divisible by 3 are less than a fixed amount and the orders are less than a fixed amount, the number of even elements (for a given number of odd elements) up to the

Properties of 3

(56) When the domain of the absolute values of the

(The

As shown previously, the number of even elements in the extended sequence is sometimes smaller than the number of odd elements for a one-jump

When

When

When

{4, 2, 1}

{24, 12, 6, 3, 10, 5, 16, 8}

{36, 18, 9, 28, 14, 7, 22, 11, 34, 17}

{30, 15, 46, 23}

{26, 13, 40, 20}

{38, 19}

{42, 21}

{32}

{44}

{25}

{27}

{29}

{31}

{33}

{35}

{37}

{39}

{41}

{43}

{45}

{47}

Each limb consists of a snippet from the 3

Note that there is no overall "gain" in a cycle. The feature of least-residue trees that makes them relevant to cycle formation is that there is no gain (at least in principle) from the end of a limb ending in an odd natural number greater than 2

For convenience, the following sets will be defined for

The limb comprising

For

When

When

The tables above show which types of limbs attach to other types of limbs and that only a limb in

(59) If a limb in

Suppose

(60) A limb in

This property further reduces the probability of a limb in

(61) If

(62) If

(63) If

(64) If

(65) If a limb in

(66) The fourth element of a limb in

(67) A four-element limb in

(68) A two-element limb in

(69) There are no limbs in

The many common properties of the least-residue trees of the 3

In this section,

length=1,

length=2,

length=4,

length=5,

length=7,

length=9,

length=10,

length=12,

length=14,

length=15,

length=17,

length=18,

Note that there are "consecutive length" pairs (1 and 2, 4 and 5, 9 and 10, 14 and 15, 17 and 18, ...) and that two times an

The first step in proving that

The formula for the

(a) The set of integers with coefficient stopping time

(b) Let

By part (b) of Terras' theorem, if

In his 1978 article, Crandall used continued fractions to show that if

More miscellaneous results pertaining to allowable (not necessarily admissible) sequence vectors and the maximum

(1, 2, 1, 2, 1, 2, 2, 1, 1)

(1, 2, 2, 1, 1, 2, 2, 1, 1)

(1, 2, 2, 1, 2, 1, 2, 1, 1)

(1, 2, 2, 1, 2, 2, 1, 1, 1)

(1, 2, 2, 2, 1, 2, 1, 1, 1)

(1, 2, 2, 2, 2, 1, 1, 1, 1)

The first sequence vector corresponds to the smallest

(1, 2, 1, 2, 2, 1, 2, 1, 2, 1)

(1, 2, 1, 2, 2, 2, 1, 1, 2, 1)

(1, 2, 1, 2, 2, 2, 1, 2, 1, 1)

(1, 2, 2, 1, 2, 2, 1, 2, 1, 1)

(1, 2, 2, 2, 1, 2, 1, 2, 1, 1)

(1, 2, 2, 2, 1, 2, 2, 1, 1, 1)

(1, 2, 2, 2, 2, 1, 2, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 1, 1, 1, 1)

In this case,

(1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1)

(1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1)

(1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1)

(1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1)

(1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1)

(1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1)

(1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1)

In this case,

(1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1)

(1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1)

(1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1)

(1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1)

(1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1)

(1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1)

(1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1)

(1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1)

(1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1)

In this case,

(1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1)

(1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1)

(1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1)

(1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1)

(1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1)

(1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1)

(1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1)

In this case,

The remaining step is to find the smallest order for which a given

4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 1 -1 0 0 0 0 0 0

9 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0

12 0 0 0 0 0 -1 1 -1 1 -1 -1 0 0 0 -1 -1 0 0 0 0 0 0

14 -1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 -1 0

17 0 1 1 1 0 1 0 1 -1 1 0 1 0 2 1 1 0 1 0 0 1 0

20 0 -1 0 -1 1 -1 1 0 1 -1 1 -1 1 0 1 -1 0 -1 1 -1 1 -1

22 0 -1 1 0 -3 -1 -2 0 -1 -1 -1 -2 -3 -2 -1 -3 -2 0 1 1 1 2

25 0 0 0 1 2 1 -2 3 0 5 4 4 1 2 0 5 0 -1 -4 1 1 6

27 0 0 0 0 1 -1 1 2 3 -1 0 4 -1 1 2 1 5 2 1 1 0 0

30 0 0 0 0 0 1 3 -3 1 0 -2 -2 -1 1 -2 -2 -2 1 0 1 1 4

33 0 0 0 0 0 0 0 0 -1 1 -1 1 0 -4 1 0 2 -3 2 -5 0 -7

For every order and limb length, the difference between the number of live limbs in

The above property can be used to estimate the smallest order for which a live limb in

When

length=12,

length=14,

length=15,

length=17,

length=18,

For a given limb length, the largest

A rotation of the parity vector ]

For limb lengths less than or equal to 51, no limb in

For a live limb in

In this section,

For the first population and prime

For the second population and prime

The minima in the second population do not increase monotonically within a period; there is a characteristic dip about half-way through the period. For example, a graph of truncated upper bounds of minima (scaled by 32768.0) for

Similarly, there is a dip about a third of the way through the period and another dip about two-thirds of the way through the period for the minima in the third population. For example, a graph of truncated upper bounds of minima (scaled by 32768.0) for

A graph of the maximum upper bounds of minima (scaled by 65536.0) for

(The scaled maximum upper bounds of minima for

A graph of the sorted maximum upper bounds of minima for

(9) J. Simons and B. de Weger,

(10) Edward G. Belaga [

(11) Georges Rhin, "Approximants de Padé et mesures effectives d'irrationalité",

(12) T. Brox, "Collatz cycles with few descents",

(13) Hardy, G. H. and Littlewood, J. E.,

(14) Edward G. Belaga, Maurice Mignotte [

(15) Mihăilescu, P.,

MSVC++™ C programs were used to confirm the above propositions. Readers may copy and modify the software in this section. No guarantee is made that it is error-free.

Use test0a to find cycles in the 3

Use test1c to histogram the |

Use test0b to regenerate a cycle in the 3

Use test0c to regenerate cycles in the 3

Use test0d to regenerate cycles in the 3

Use test0e to regenerate cycles in the 3

list consists of the cycles found for

list1 consists of one-attachment-point cycles found for

Use test1 to find the number of jumps in the 3

Use test2 to generate a histogram of limb lengths for a 3

Use test3a to generate limbs of least-residue trees where at least one element of the limb is divisible by 8.

Use test3b to generate limbs of least-residue trees where at least one element of the limb is divisible by 8. Multiple-word arithmetic is used.

Use test3c to generate limbs of least-residue trees where at least one element of the limb is divisible by 8. Multiple-word arithmetic is used. This C program is for use on the TMS320C64™ digital signal processing chip (with hand-optimized assembly language subroutines).

Use test4a12, test4a13, test4a14, test4a15, test4a16, test4a17, test4a18, test4a19, test4a20, test4a21, test4a22, test4a23, test4a24, test4a25, test4a26, test4a27, test4a28, test4a29, test4a30, test4a31, test4a32, and test4a33 to compare the number of limbs in

Use test5a to generate tumbles and jumps having three peaks and two troughs.

Use test5b to generate tumbles and jumps having four peaks and three troughs.

Use test6a to compute the largest

Use test6b to compute the largest

Use test7 to check that

Use test8 to check that

Use test9 to generate least-residue trees. The permutation of

Use test10 to generate least-residue trees for the 3

Use test11 to generate long limbs in

Use test12 to find sequence vectors corresponding to different

Use test13 to compute

Use test14a to compute the minimum element in a cycle using the formula

Use test15 to generate parity vectors (distinct under rotation) for given

Use test16 to determine that the (

Multiple-word arithmetic TMS320C64™ assembly language subroutines are add64, div12864, div6432, mul6432, mul6464, mul3232, shift64, sub64, subr2, and n-word arithmetic.

In his book

Generalized Fibonacci and Lucas series are discussed in the link nroots.

The cross-ratio function is discussed in the link cross.

The Farey series is discussed in the link farey.

The relationship between the Farey series and the Riemann hypothesis is discussed in the link riemann.