OEIS/Collatz Story
Contents
- 1 Abstract
- 2 Introduction
- 3 Collatz Graph
- 4 Segments
- 5 Segment Tree
- 5.1 Attachment Directory Construction
- 5.2 Attachment rules
- 5.3 Properties of the Attachment Rules
- 5.4 Shifting left or right
- 5.5 Decreasing and increasing set of subtrees
- 5.6 Reduction of the increasing set
- 5.7 Rule sieving
- 5.8 Order of Rule Application
- 5.9 Attachment Process
- 5.10 Supersegments
- 5.11 Distribution of Supernodes
- 5.12 Attachment of Segments with Supernodes
- 5.13 Degree 2
- 5.14 Degree 3
- 5.15 Degree 4
- 5.16 No Cycles
- 6 Proof for the Collatz Tree
- 7 Introduction
- 8 Collatz Graph
- 9 Segments
- 10 Segment Tree
- 10.1 Attachment Directory Construction
- 10.2 Attachment rules
- 10.3 Properties of the Attachment Rules
- 10.4 Shifting left or right
- 10.5 Decreasing and increasing set of subtrees
- 10.6 Reduction of the increasing set
- 10.7 Rule sieving
- 10.8 Order of Rule Application
- 10.9 Attachment Process
- 10.10 Supersegments
- 10.11 No Cycles
- 11 Proof for the Collatz Tree
Abstract
Small, finite trees with two branches are constructed with the operations defined by Collatz for his 3x+1 problem. These trees are connected to form bigger graphs in an iterative process. It is shown that this process finally builds a single graph which is a tree except for one cycle at the root. This graph is then embedded into the Collatz graph, and it is thereby shown that the latter is also a tree except for the cycle 4-2-1.
Introduction
Collatz sequences (also called trajectories) are sequences of integer numbers > 0. For any start value > 0 the elements of the sequence are constructed with two simple rules:
- Even numbers are halved.
- Odd numbers are multiplied by 3 and then incremented by 1.
Since decades it is unknown whether the final cyle 4 - 2 - 1 is always reached for all start values. This problem is the Collatz conjecture, for which the English Wikipedia states:
- It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.
Simple visualizations of Collatz sequences show no obvious structure. The sequences for the first dozen of start values are rather short, but the sequence for 27 suddenly has 112 elements.
Da sieht man den Wald vor lauter Bämen nicht.
German proverb: You cannot see the wood for the trees.
References
- Jeffry C. Lagarias, Ed.: The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. MBK78
- OEIS A07165: File of first 10K Collatz sequences, ascending start values, with lengths
- Manfred Trümper: The Collatz Problem in the Light of an Infinite Free Semigroup. Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 p.
Collatz Graph
When all Collatz sequences are read backwards, they form the Collatz graph starting with 1, 2, 4, 8 ... . At each node m > 4 in the graph, the path from the root (4) can be continued
- always to m * 2, and
- to (m - 1) / 3 if m ≡ 1 mod 3.
The Collatz conjecture claims that the graphs contains all numbers, and that - except for the leading cycle 1 - 2 - 4 - 1 - 2 - 4 ... - it has the form of a tree (without cycles). We will not consider the leading cycle, and we start the graph with node 4, the root. Furthermore, another trivial type of path starts when m ≡ 0 mod 3. We call such a path a sprout, and it contains duplications only. Sprouts must be added to the graph for any node divisible by 3, therefore we will not consider them for the moment.
Graph Operations
Following Trümper, we use abbreviations for the elementary operations which transform a node (element, number) in the Collatz graph according to the following table (T1):
Name | Mnemonic | Distance to root | Mapping | Condition |
---|---|---|---|---|
d | down | -1 | m ↦ m / 2 | m ≡ 0 mod 2 |
u | up | -1 | m ↦ 3 * m + 1 | (m ≡ 1 mod 2) |
s := ud | spike | -2 | m ↦ (3 * m + 1) / 2) | m ≡ 1 mod 2 |
δ | divide | +1 | m ↦ (m - 1) / 3 | m ≡ 1 mod 3 |
µ | multiply | +1 | m ↦ m * 2 | (none) |
σ := δµ | squeeze | +2 | m ↦ ((m - 1) / 3) * 2 | m ≡ 1 mod 3 |
We will mainly be interested in the reverse mappings (denoted with greek letters) which move away from the root of the graph.
3-by-2 Replacement
The σ operation, applied to numbers of the form 6 * m - 2, has an interesting property:
(6 * (3 * n) - 2) σ = 4 * 3 * n - 2 = 6 * (2 * n) - 2
In other words, as long as m contains a factor 3, the σ operation maintains the form 6 * x - 2, and it replaces the factor 3 by 2 (it "squeezes" a 3 into a 2). In the opposite direction, the s operation replaces a factor 2 in m by 3.
Motivation: Patterns in sequences with the same length
A closer look at the Collatz sequences shows a lot of pairs of adjacent start values which have the same sequence length, for example (from OEIS A070165):
142/104: 142 d 71 u 214 d 107 u 322 d 161 u 484 d 242 d 121 u 364 ] 182, 91, ... 4, 2, 1 143/104: 143 u 430 d 215 u 646 d 323 u 970 d 485 u 1456 d 728 d 364 ] 182, 91, ... 4, 2, 1 +1 *6+4 +1 *6+4 +1 *6+4 +1 *6+4 *6+2 +0 +0 ...
The third line tells how the second line could be computed from the first. Proceeding from right to left, the step pattern is:
δ µ µ δ µ δ µ δ µ µ µ δ µ δ µ δ µ δ
The alternating pattern of operations can be continued to the left with 4 additional pairs of steps:
q? u [ 62 d 31 u 94 d 47 u 142 d ... 126 d [ 63 u 190 d 95 u 286 d 143 u ... +1 *6+4 +1 *6+4 +1
The pattern stops here since there is no number q such that q * 3 + 1 = 62.
Segments
These patterns lead us to the construction of special subsets of paths in the Collatz graph which we call segments. They lead away from the root, and they always start with a node m ≡ -2 mod 6. Then they split and follow two subpaths in a prescribed sequence of operations. The segment construction process is stopped when the next node in one of the two subpaths becomes divisible by 3, resp. when a δ operation is no more possible.
Segment Directory Construction
We list the segments as rows of an infinite array C[i,j], the so-called segment directory.
- Informally, and in the two examples above, we consider the terms betweeen the square brackets. For the moment, we only take those which are which are ≡ 4 mod 6 (for "compressed" segments; below there are also "detailed" segments where we take all). We start at the right and with the lower line, and we interleave the terms ≡ 4 mod 6 of the two lines to get a segment.
Continuing the example above:
[ 62 d 31 u 94 d 47 u 142 d 71 u 214 d 107 u 322 d 161 u 484 d 242 d 121 u 364 ] [ 63 u 190 d 95 u 286 d 143 u 430 d 215 u 646 d 323 u 970 d 485 u 1456 d 728 d 364 ]
Left-to-right reversed, only terms of the form 6*m+4, rows switched and without operations:
364 1456 970 644 430 286 190 364 484 322 214 142 94
The final, linearized example segment in row 61 of the directory looks like:
61 | 364 | 1456 | 484 | 970 | 322 | 646 | 214 | 430 | 142 | 286 | 94 | 190 |
The first column(s) C[i,1] will be denoted as the left side of the segments (or of the whole directory), while the columns C[i,j], j > 1 are the right part.
The following table (T2) tells how the columns j in one row i of C must be constructed if the condition is fulfilled:
Column j | Operation | Formula | Condition | Sequence |
---|---|---|---|---|
1 | C[i,1] | 6 * i - 2 | 4, 10, 16, 22, 28, ... | |
2 | C[i,1] µµ | 24 * (i - 1) / 1 + 16 | 16, 40, 64, 88, 112, ... | |
3 | C[i,1] δµµ | 24 * (i - 1) / 3 + 4 | i ≡ 1 mod 3 | 4, 28, 52, 76, 100, ... |
4 | C[i,2] σ | 48 * (i - 1) / 3 + 10 | i ≡ 1 mod 3 | 10, 58, 106, 134, ... |
5 | C[i,3] σ | 48 * (i - 7) / 9 + 34 | i ≡ 7 mod 9 | 34, 82, 130, 178, ... |
6 | C[i,4] σ | 96 * (i - 7) / 9 + 70 | i ≡ 7 mod 9 | 70, 166, 262, 358, ... |
7 | C[i,5] σ | 96 * (i - 7) / 27 + 22 | i ≡ 7 mod 27 | 22, 118, 214, 310, ... |
8 | C[i,6] σ | 192 * (i - 7) / 27 + 46 | i ≡ 7 mod 27 | 46, 238, 430, 622, ... |
9 | C[i,7] σ | 192 * (i - 61) / 81 + 142 | i ≡ 61 mod 81 | 142, 334, ... |
... | ... | ... | ... | ... |
j | C[i,j-2] σ | 6 * 2^{k+1} * (i - m) / 3^{l} + 3 * 2^{k} * h - 2 | i ≡ m mod 3^{l} | ... |
The general formula for a column j >= 4 uses the following parameters:
- k = floor(j / 2)
- l = floor(j - 1) / 2)
- m = a(floor((j - 1) / 4), where a(n) is the OEIS sequence (A066443: a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0 . The values are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides (4), 40, 364, 3280, 29524 (OEIS A191681). The constants appear first in columns 2-4 (in segment 1), 5-8 (in segment 7), 9-12 (in segment 61) and so on
- h = a(j), where a(n) is the OEIS sequence A084101 with period 4: a(0..3) = 1, 3, 3, 1; a(n) = a(n - 4) for n > 3.
(This results in k = 2, l = 1, m = 1, h = 1 for j = 4.)
The first few lines of the segment directory are the following:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ... | 2*j | 2*j+1 | |
i | 6*i‑2 | µµ | δµµ | µµσ | δµµσ | µµσσ | δµµσσ | µµσ^{3} | δµµσ^{3} | µµσ^{4} | δµµσ^{4} | ... | µµσ^{j-1} | δµµσ^{j-1} |
1 | 4 | 16 | 4 | 10 | ||||||||||
2 | 10 | 40 | ||||||||||||
3 | 16 | 64 | ||||||||||||
4 | 22 | 88 | 28 | 58 | ||||||||||
5 | 28 | 112 | ||||||||||||
6 | 34 | 136 | ||||||||||||
7 | 40 | 160 | 52 | 106 | 34 | 70 | 22 | 46 |
There is a more elaborated segment directory with 5000 rows.
Properties of the Segment Directory
We make a number of claims for the segment directory C:
- (C1) All nodes in the segment directory have the form 6 * n - 2.
- This follows from the formula for columns C[i,1..3], and for any higher column numbers from the 3-by-2 replacement property of the σ operation.
- (C2) All segments have a finite length.
- At some point the σ operations will have replaced all factors 3 by 2.
- (C3) All nodes in the right part of a segment have the form 6 * (3^{n} * 2^{m} * f) - 2 with the same "3-2-free" factor f.
- This follows from the operations for columns C[i,1..3], and from the fact that the σ operation maintains this property.
- (C4) All nodes in the right part of a particular segment are
- different among themselves, and
- different from the left side of that segment (except for the first segment for the root 4).
- For C[i,1..2] we see that the values modulo 24 are different. For the remaining columns, we see that the exponents of the factors 2 and 3 are different. They are shifted by the σ operations, but they alternate, for example (in the segment with left part 40):
160 = 6 * (3^{3} * 2^{0} * 1) - 2 52 = 6 * (3^{2} * 2^{0} * 1) - 2 106 = 6 * (3^{2} * 2^{1} * 1) - 2 34 = 6 * (3^{1} * 2^{1} * 1) - 2 70 = 6 * (3^{1} * 2^{2} * 1) - 2 22 = 6 * (3^{0} * 2^{2} * 1) - 2 46 = 6 * (3^{0} * 2^{3} * 1) - 2
- (C5) There is no cycle in a segment (except for the first segment for the root 4).
Segment Lengths
Oviously the segment directory is very structured. The lengths of the compressed segments follow the pattern
4 2 2 4 2 2 L_{1} 2 2 4 2 2 4 2 2 L_{2} 2 2 4 2 2 ...
with two fixed lengths 2 and 4 and some variable lengths L_{1}, L_{2} ... > 4. For the left parts 4, 40, 364, 3280, 29524 (OEIS A191681), the segment lengths have high values 4, 8, 12, 16, 20 which did not occur before. Those left parts are (9^{n+1} - 1) / 2, or 4 * Sum(9^{i}, i = 0..n).
Coverage of the Right Part
We now examine the modular conditions which result from the segment construction table (T2) in order to find out how the numbers of the form 6 * n - 2 are covered by the right part of the segment directory. The following table (T3) shows the result:
Columns j | Covered | Remaining |
---|---|---|
2-3 | 4, 16 mod 24 | 10, 22, 34, 46 mod 48 |
3-4 | 10, 34 mod 48 | 22, 46, 70, 94 mod 96 |
5-6 | 70, 22 mod 96 | 46, 94, 142, 190 mod 192 |
7-8 | 46, 142 mod 192 | 94, 190, 286, 382 mod 384 |
... | ... | ... |
We can always exclude the first and the third element remaining so far by looking in the next two columns of segments with sufficient length.
- (C6) There is no limit on the length of a segment.
- We only need to take a segment which, in its right part, has a factor of 3 with a sufficiently high power, and the σ operations will stretch out the segment accordingly.
Therefore we can continue the modulus table above indefinitely, which leads us to the claim:
- (C7) All numbers of the form 6 * n - 2 occur exactly once in the right part of the segment directory, and once as a left side. There is a bijective mapping between the left sides and the elements of the right parts.
- The sequences defined by the columns in the right part all have different modulus conditions. Therefore they are all disjoint. The left sides are disjoint by construction.
Segment Tree
So far we possess the segment directory C which represents the root segment and an infinite set of small trees with disjoint nodes and two branches. We know that the segments represent trees, and that their right parts are all disjoint and different from the left side.
We now want to attach (or connect) the segments to other graphs until we get a single big graph which will later become the backbone of the Collatz graph. Ideally the attachment process should maintain the tree property of the graphs all the time.
- The verb attach emphasizes the direction of the operation better than the verb connect.
Attachment Directory Construction
Parallel to the segment directory we maintain the attachment directory A which, for any source segment in C:
- tells whether the tree corresponding to the segment was already attached to the graph represented by some other segment, and if so,
- tells the target segment and column numbers in the segment directory C where the source segment was attached.
Initially all segments are unattached.
We operate on A as follows: Considering simultaneously a set of source segments i > 1 (i.e. omitting the root segment) in C - which fulfill some modularity condition (the source segment set), and which are so far unattached, we attach their segments parallel to the unique occurrences of their left sides in the right part of C (target segment set and target column).
Attachment rules
The following table (T4) tells the computation rules for the target position, depending on the modularity condition of the source segment. We identify and denote these attachment rules by the target column number. We show the the first segments (their left side) for k = 0, 1, 2, 3.
Rule / column |
Source segments |
Condition / remaining |
First source segments |
Target segments |
First target segments |
Dir. |
---|---|---|---|---|---|---|
5 | 6(2^{0}(4k + 3)) - 2 | 0 mod 8 2, 4, 6 mod 8 |
16, 40, 64, 88 | 6(3^{0}k + 1 ) - 2 | 4, 10, 16, 22 | < |
6 | 6(2^{0}(4k + 1)) - 2 | 4 mod 8 2, 6, 10, 14 mod 16 |
4, 28, 52, 76 | 6(3^{1}k + 1) - 2 | 4, 22, 40, 58 | < |
9 | 6(2^{1}(4k + 1)) - 2 | 10 mod 16 2, 6, 14 mod 16 |
10, 58, 106, 154 | 6(3^{1}k + 1) - 2 | 4, 22, 40, 58 | < |
10 | 6(2^{1}(4k + 3)) - 2 | 2 mod 16 6, 14, 22, 30 mod 32 |
34, 82, 130, 178 | 6(3^{2}k + 7) - 2 | 40, 94, 148, 202 | > |
13 | 6(2^{2}(4k + 3)) - 2 | 6 mod 32 14, 22, 30 mod 32 |
70, 166, 262, 358 | 6(3^{2}k + 7) - 2 | 40, 94, 148, 202 | < |
14 | 6(2^{2}(4k + 1)) - 2 | 22 mod 32 14, 30, 46, 62 mod 64 |
22, 118, 214, 310 | 6(3^{3}k + 7) - 2 | 40, 202, 364, 526 | > |
17 | 6(2^{3}(4k + 1)) - 2 | 46 mod 64 14, 30, 62 mod 64 |
46, 238, 430, 622 | 6(3^{3}k + 7) - 2 | 40, 202, 364, 526 | < |
18 | 6(2^{3}(4k + 3)) - 2 | 14 mod 64 30, 62, 94, 126 mod 128 |
142, 334, 526, 718 | 6(3^{4}k + 61) - 2 | 364, 850, 1336, 1822 | > |
21 | 6(2^{4}(4k + 3)) - 2 | 30 mod 128 62, 94, 126 mod 128 |
286, 670, 1054, 1438 | 6(3^{4}k + 61) - 2 | 364, 850, 1336, 1822 | > |
22 | 6(2^{4}(4k + 1)) - 2 | 94 mod 128 62, 126, 190, 254 mod 256 |
94, 478, 862, 1246 | 6(3^{5}k + 61) - 2 | 364, 1822, 3280, 4738 | > |
... | ... | ... | ... | ... | > | ... |
It should be obvious how the following rows of the table must be filled. The additive constants in the formula for the source segments follow the periodic pattern 3, 1, 1, 3 (OEIS A084101), while those for the target segments are taken from OEIS A066443. The latter constants change in every fourth row of (T4).
As an example, we apply rule 14 to source segment 22. (This example does not show the result of of the whole process, but only a single step.)
1 | 5 | 6 | 9 | 10 | 13 | 14 | 17 | 18 | 21 | 22 | ... | |
1 | 4 | 16 | 4 | 10 | ||||||||
2 | 10 | 40 | ||||||||||
3 | 16 | 64 | ||||||||||
4 | ||||||||||||
5 | 28 | 112 | ||||||||||
6 | 34 | 136 | ||||||||||
7 | 40 | 160 | 52 | 106 | 34 | 70 | 22 | 46 | ||||
22 | 88 | 28 | 58 |
Properties of the Attachment Rules
For the attachment directory A we note respectively claim:
- (A1) The source segments met by the conditions in the rules are all disjoint.
- (A2) Therefore, a source segment is chosen by the process exactly once.
- (A3) Each source segment meets a condition in some rule with a sufficiently high number.
- (A4) The construction is such that the target column always exists in the target segments.
- Table (T4) is derived from (T2) which has similiar modularity conditions.
- (A5) The target column (or rule number) depends on the modularity condition for the source segment alone, but not on (the left side of) the segment.
- This can be shown by the graph operations (δ / µ / σ) which are tied to the columns.
Shifting left or right
There are two categories of attachment rules (column Dir. in (T4)):
- Rules 5, 6, 9, 13, 17 attach to lower segments - they shift left.
- Rules 10, 14, 18 and above attach to higher segments - they shift right.
- This can be seen from the powers of 2 and 3 in the source and target row columns. Starting at segment 18, we have 3^{k} > 2^{k+2} for k >= 4.
With the single exception of the root segment 1, the rules obviously never attach a segment to itself.
Decreasing and increasing set of subtrees
Likewise, we can also group the subtrees which are built from the segments by attachment operations into two sets:
- the decreasing set T_{d} with members that will attach to some segment with a lower number, initially the segments for which rules 5, 6, 9, 13, 17 apply.
- the increasing set T_{i} with members that will attach to some segment with a higher number, initially the segments for which rules 10, 14, 18 and above apply.
We define that the root segment is also a member of T_{d}. The goal is the following claim: (A7) If T_{i} is empty, all segments in T_{d} finally attach to the tree above the root segment.
- Suppose n is the smallest member of T_{d} which is not yet attached. A left-shifting rule applies to n, but there is no smaller, unattached member in T_{d}. T_{i} is empty. Therefore n must be attached to the root or some segment in the tree above the root.
Reduction of the increasing set
We now try to move subsets of T_{i} to T_{d} by examining the parameter k in the formula for the targets t of some members s in T_{i} (c.f. (T4)). We concentrate on rule 10 because the targets of rules 14, 18 and above are a subset of the targets of rule 10 (i.e. the "longer" segments 4, 22, 40, 58, 76, 94 ...).
A simple observation is:
- (A8) We can move all members with even k.
- We attach s to t. For even k = 2l we get an odd factor of 6: t = 6*(2l + 1) - 2 = 12l + 4 which implies t ≡ 0, 4 mod 8 with the left-shifting rules 5 or 6. Therefore t (and the attached s) can be moved to T_{d}.
We can now assume that T_{i} contains only segments with odd k in the target formula.
- (A9) If k = 2l + 1 is odd, then t is a supersegment of degree >= 2.
- The constants 1, 7, 61 ... from OEIS A066443 have the formula a(n) = 1 + Sum_{i=1..n} 2*3^(2i-1), n >= 0, which implies a(n) ≡ 1 mod 6. We have:
t = 6(3^{m}(2l + 1) + 1 + 6j) - 2 = 6(6*3^{m-1}*l + 3^{m} + 1 + 6j) - 2 = 6(6*(3^{m-1}*l + j) + 3^{m} + 1) - 2
- 3^{m} + 1 ≡ 4 mod 6 can be proven by induction, therefore t has the form 6(6i - 2) - 2 of a supernode.
So we are left with the task to examine the supersegments for which right-shifting rules apply.
Rule sieving
Order of Rule Application
- (A7) The resulting graphs do not depend on the order of application of the attachment rules.
- The rules may well hit the same target segments, but they always do so in different target columns. It does not matter whether the target segment is already attached.
Despite of (A6) we will apply the rules in a well-defined order, because only in this order we can show that the tree property of the subgraphs is always maintained.
Attachment Process
We will now use the rules of (T4) to reduce the set of unattached segments in C in an iterative process. Our goal is to show that all segments are attached - mostly indirectly - to the root segment.
Supersegments
The segments considered so far contain nodes of the form 6 * i - 2. We call a node where i has the same form a supernode (of degree 2, 3, 4 and so on):
n_{1} = 6 * i - 2 = 6 * i - 2 ≡ 0 mod 2 n_{2} = 6 * (6 * i - 2) - 2 = 36 * i - 14 ≡ 2 mod 4: rules >= 9 n_{3} = 6 * (6 * (6 * i - 2) - 2) - 2 = 216 * i - 86 ≡ 2 mod 8; rules 9, 10 n_{4} = ... = 1296 * i - 518 ≡ 10 mod 16; rule 9 n_{5} = ... = 7776 * i - 3110 ≡ 10 mod 16; rule 9 ... n_{j} = 6^{j} * i - m_{j}
The additive constants m_{j} are taken here from OEIS A005610 with a(k) = 6 * a(k - 1) + 2 = 2 * (6 * 6^{k} - 1) / 5.
When a segment has a supernode as its left side, it is called a supersegment. An inspection of the segment directory C shows that supernodes occur at the following source positions (table (T5)):
Degree | Column | First source rows | Difference |
---|---|---|---|
2 | 1 | 4, 10, 16, 22 ... | 6^{1} |
9 < | 4, 13, 22, 31 ... | 9 | |
10 > | 25, 52, 79, 106 ... | 27 | |
13 < | 16, 43, 70, 97 ... | 27 | |
... | ... | ... | |
3 | 1 | 22, 58, 94, 130 ... | 6^{2} |
9 < | 22, 49, 76, 103 ... | 27 | |
10 > | 25, 106, 187, 268 ... | 81 | |
4 | 1 | 130, 346, 562, 778 ... | 6^{3} |
9 < | 49, 130, 211, 292 ... | 81 | |
5 | 1 | 778, 2074, 3370 ... | 6^{4} |
9 < | 292, 778, 1264, 1750, 2236 ... | 486 = 6 * 81 |
That are a rather simple consequences of the segment construction rules. We state some claims which are not so obvious:
- (S1) For degrees > 2, no other columns than the ones shown in table (T5) are occupied by supernodes of that degree.
- (S2) For degrees >= 4, only rule 9 (which moves downwards) is applicable.
- The property ≡ 10 mod 16 is maintained by the map i => 6 * i - 2 because 6 * 10 - 2 = 58 ≡ 10 mod 16.
- (S3) Supernodes only occur in segments s ≡ 4 mod 18. (These are the segments which have at least 6 columns).
- (S4) There is not more than one supernode in the right part of a segment, and if there is one, it occurs at the last or the last-but-one position in the right part (which represent the leafs of the corresponding trees).
- (S5) Each segment which contains a supernode in its right part:
- either has an odd row number,
- or a supernode as its left side.
- (S6) Each segment which does not contain a supernode in its right part (that are rows 1, 10, 19, 28, 37, 46, 55 ... i ≡ 1 mod 9):
- either has an odd row number,
- or a supernode as its left side.
We first attach all even rows mentioned in (S3). Then we attach the even rows mentioned in (S4).
Distribution of Supernodes
- (S??) Suppernodes occur only in the "longer" segments with i ≡ 1 mod 3.
- (S??) If there is a supernode in the right part of a segment, it is either at the end, i.e. in the last or the last-but-one column.
- (S??) There are at most two supernodes in every segment.
Attachment of Segments with Supernodes
It is obvious that the supernodes inherit the properties of the nodes 6 * i - 2:
- (S??) Supernodes occur exactly once as a left side and in the right part of the segment directory C.
Degree 2
- (S??) There are repeating blocks of rows i = 18m + 1 + 3j, m = 0..., j = 0..5 with the following pattern (table (T??)):
j | i for m = 0 |
Rule | Degree Column 1 |
Degree Column 9 |
Degree last(-1) Column >= 10 |
---|---|---|---|---|---|
0 | 1 | 5/6 left | |||
1 | 4 | left/right | 2+ | 2+ | |
2 | 7 | 6/5 left | 2 | ||
3 | 10 | left/right | 2 | ||
4 | 13 | 5/6 left | 2 | ||
5 | 16 | left/right | 2 | 2 |
This implies:
- (S??) Segments with odd i are always shifted left.
- (S??) Segments where column 9 has degree 4 are either shifted left, or their left side has a degree 2 or 3.
Degree 3
- (S??) Degree 2 occurs in columns 1, 9 and 10 only.
- (S??) For columns 1 and 9, there are repeating blocks of rows i = 108m + 4 + 9j, m = 0..., j = 0..11 with the following pattern (table (T??)):
j | i for m = 0 |
Rule | Degree Column 1 |
Degree Column 9 |
---|---|---|---|---|
0 | 4 | left/right | 2 | 2 |
1 | 13 | 6 left | 2 | |
2 | 22 | left/right | 3+ | 3+ |
3 | 31 | 5 left | 2 | |
4 | 40 | left/right | 2 | 2 |
5 | 49 | 6 left | 3+ | |
6 | 58 | left/right | 3 | 2 |
7 | 67 | 5 left | 2 | |
8 | 76 | left/right | 2 | 3+ |
9 | 85 | 6 left | 2 | |
10 | 94 | left/right | 3 | 2 |
11 | 103 | 5 left | 3+ |
Furthermroe, there are segments where column 10 has degree 3 and column 1 has degree 1 or 2, namely in rows i = 81m + 25.
Degree 4
- (S??) Degree 4 occurs in columns 1 and 9 only.
- (S??) There are repeating blocks of rows i = 648m + 49 + 81j, m = 0..., j = 0..7 with the following pattern (table (T??)):
j | i for m = 0 |
Rule | Degree Column 1 |
Degree Column 9 |
---|---|---|---|---|
0 | 49 | 6 left | 4+ | |
1 | 130 | 9 left | 4+ | 4+ |
2 | 211 | 5 left | 4+ | |
3 | 292 | right | 2 | 4+ |
4 | 373 | 6 left | 4+ | |
5 | 454 | right | 3 | 4+ |
6 | 535 | 5 left | 4+ | |
7 | 616 | left/right | 2 | 4+ |
This implies:
- (S??) Segments where column 1 (the left side, j = 1) has degree 4 are always shifted left.
- (S??) Segments where column 9 has degree 4 are either shifted left, or their left side has a degree 2 or 3.
- (S??) All nodes with degree 4 in column 9 occur in left-shifting segments except for rows 454, 1102*, 1750, 2398, 3046, 3694, 4342, 4990* ... (delta 648). Of these, all except the ones with "*" (every 6th) shift left for the target. If the left side has degree 4, then the segment is shifted left.
- (S??) All segments with a left side of degree 4 shift left and can be moved into the low forest.
No Cycles
- (A8) The attachment process does not create any new cycle (in addition to the one in the root segment).
- Let a segment/tree t_{1} with left side n_{1} and right part R_{1} be attached to node n_{1} in the right part R_{2} of the unique segment/tree t_{2} which has the left side by n_{2}. t_{1} and t_{2} are disjoint trees by (C4), therefore the result of such a single attachment step is a tree again (u_{2}, still with left side n_{2}).
Proof for the Collatz Tree
- (P1) The remaining single tree is a subgraph of the Collatz graph.
- The edges of the compressed tree carry combined operations µµ, δµµ and σ = δµ.
So far, numbers of the form x ≡ 0, 1, 2, 3, 5 mod 6 are missing from the compressed tree.
We insert intermediate nodes into the compressed tree by applying operations on the left parts of the segments as shown in the following table (T5):
Operation | Condition | Resulting Nodes | Remaining Nodes |
---|---|---|---|
δ | 2 * i - 1 | i ≡ 0, 2, 6, 8 mod 12 | |
µ | 12 * i - 4 | i ≡ 0, 2, 6 mod 12 | |
δµ | i ≡ 1, 2 mod 3 | 4 * i - 2 | i ≡ 0, 12 mod 24 |
δµµ | i ≡ 2 mod 3 | 8 * i - 4 | i ≡ 0 mod 24 |
δµµµ | i ≡ 2 mod 3 | 16 * i - 8 | (none) |
The first three rows in T5 care for the intermediate nodes at the beginning of the segment construction with columns 1, 2, 3. Rows 4 and 5 generate the sprouts (starting at multiples of 3) which are not contained in the segment directory.
We call such a construction a detailed segment (in contrast to the compressed segments described above).
- A detailed segment directory can be created by the same Perl program. In that directory, the two subpaths of a segment are shown in two lines. Only the highlighted nodes are unique.
- (P2) The connectivity of the compressed tree remains unaffected by the insertions.
- (P3) With the insertions of (T5), the compressed tree covers the whole Collatz graph.
- (P4) The Collatz graph is a tree (except for the cycle 4-2-1.
Introduction
Collatz sequences (also called trajectories) are sequences of integer numbers > 0. For any start value > 0 the elements of the sequence are constructed with two simple rules:
- Even numbers are halved.
- Odd numbers are multiplied by 3 and then incremented by 1.
Since decades it is unknown whether the final cyle 4 - 2 - 1 is always reached for all start values. This problem is the Collatz conjecture, for which the English Wikipedia states:
- It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.
Simple visualizations of Collatz sequences show no obvious structure. The sequences for the first dozen of start values are rather short, but the sequence for 27 suddenly has 112 elements.
Da sieht man den Wald vor lauter Bämen nicht.
German proverb: You cannot see the wood for the trees.
References
- Jeffry C. Lagarias, Ed.: The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. MBK78
- OEIS A07165: File of first 10K Collatz sequences, ascending start values, with lengths
- Manfred Trümper: The Collatz Problem in the Light of an Infinite Free Semigroup. Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 p.
Collatz Graph
When all Collatz sequences are read backwards, they form the Collatz graph starting with 1, 2, 4, 8 ... . At each node m > 4 in the graph, the path from the root (4) can be continued
- always to m * 2, and
- to (m - 1) / 3 if m ≡ 1 mod 3.
The Collatz conjecture claims that the graphs contains all numbers, and that - except for the leading cycle 1 - 2 - 4 - 1 - 2 - 4 ... - it has the form of a tree (without cycles). We will not consider the leading cycle, and we start the graph with node 4, the root. Furthermore, another trivial type of path starts when m ≡ 0 mod 3. We call such a path a sprout, and it contains duplications only. Sprouts must be added to the graph for any node divisible by 3, therefore we will not consider them for the moment.
Graph Operations
Following Trümper, we use abbreviations for the elementary operations which transform a node (element, number) in the Collatz graph according to the following table (T1):
Name | Mnemonic | Distance to root | Mapping | Condition |
---|---|---|---|---|
d | down | -1 | m ↦ m / 2 | m ≡ 0 mod 2 |
u | up | -1 | m ↦ 3 * m + 1 | (m ≡ 1 mod 2) |
s := ud | spike | -2 | m ↦ (3 * m + 1) / 2) | m ≡ 1 mod 2 |
δ | divide | +1 | m ↦ (m - 1) / 3 | m ≡ 1 mod 3 |
µ | multiply | +1 | m ↦ m * 2 | (none) |
σ := δµ | squeeze | +2 | m ↦ ((m - 1) / 3) * 2 | m ≡ 1 mod 3 |
We will mainly be interested in the reverse mappings (denoted with greek letters) which move away from the root of the graph.
3-by-2 Replacement
The σ operation, applied to numbers of the form 6 * m - 2, has an interesting property:
(6 * (3 * n) - 2) σ = 4 * 3 * n - 2 = 6 * (2 * n) - 2
In other words, as long as m contains a factor 3, the σ operation maintains the form 6 * x - 2, and it replaces the factor 3 by 2 (it "squeezes" a 3 into a 2). In the opposite direction, the s operation replaces a factor 2 in m by 3.
Motivation: Patterns in sequences with the same length
A closer look at the Collatz sequences shows a lot of pairs of adjacent start values which have the same sequence length, for example (from OEIS A070165):
142/104: 142 d 71 u 214 d 107 u 322 d 161 u 484 d 242 d 121 u 364 ] 182, 91, ... 4, 2, 1 143/104: 143 u 430 d 215 u 646 d 323 u 970 d 485 u 1456 d 728 d 364 ] 182, 91, ... 4, 2, 1 +1 *6+4 +1 *6+4 +1 *6+4 +1 *6+4 *6+2 +0 +0 ...
The third line tells how the second line could be computed from the first. Proceeding from right to left, the step pattern is:
δ µ µ δ µ δ µ δ µ µ µ δ µ δ µ δ µ δ
The alternating pattern of operations can be continued to the left with 4 additional pairs of steps:
q? u [ 62 d 31 u 94 d 47 u 142 d ... 126 d [ 63 u 190 d 95 u 286 d 143 u ... +1 *6+4 +1 *6+4 +1
The pattern stops here since there is no number q such that q * 3 + 1 = 62.
Segments
These patterns lead us to the construction of special subsets of paths in the Collatz graph which we call segments. They lead away from the root, and they always start with a node m ≡ -2 mod 6. Then they split and follow two subpaths in a prescribed sequence of operations. The segment construction process is stopped when the next node in one of the two subpaths becomes divisible by 3, resp. when a δ operation is no more possible.
Segment Directory Construction
We list the segments as rows of an infinite array C[i,j], the so-called segment directory.
- Informally, and in the two examples above, we consider the terms betweeen the square brackets. For the moment, we only take those which are which are ≡ 4 mod 6 (for "compressed" segments; below there are also "detailed" segments where we take all). We start at the right and with the lower line, and we interleave the terms ≡ 4 mod 6 of the two lines to get a segment.
Continuing the example above:
[ 62 d 31 u 94 d 47 u 142 d 71 u 214 d 107 u 322 d 161 u 484 d 242 d 121 u 364 ] [ 63 u 190 d 95 u 286 d 143 u 430 d 215 u 646 d 323 u 970 d 485 u 1456 d 728 d 364 ]
Left-to-right reversed, only terms of the form 6*m+4, rows switched and without operations:
364 1456 970 644 430 286 190 364 484 322 214 142 94
The final, linearized example segment in row 61 of the directory looks like:
61 | 364 | 1456 | 484 | 970 | 322 | 646 | 214 | 430 | 142 | 286 | 94 | 190 |
The first column(s) C[i,1] will be denoted as the left side of the segments (or of the whole directory), while the columns C[i,j], j > 1 are the right part.
The following table (T2) tells how the columns j in one row i of C must be constructed if the condition is fulfilled:
Column j | Operation | Formula | Condition | Sequence |
---|---|---|---|---|
1 | C[i,1] | 6 * i - 2 | 4, 10, 16, 22, 28, ... | |
2 | C[i,1] µµ | 24 * (i - 1) / 1 + 16 | 16, 40, 64, 88, 112, ... | |
3 | C[i,1] δµµ | 24 * (i - 1) / 3 + 4 | i ≡ 1 mod 3 | 4, 28, 52, 76, 100, ... |
4 | C[i,2] σ | 48 * (i - 1) / 3 + 10 | i ≡ 1 mod 3 | 10, 58, 106, 134, ... |
5 | C[i,3] σ | 48 * (i - 7) / 9 + 34 | i ≡ 7 mod 9 | 34, 82, 130, 178, ... |
6 | C[i,4] σ | 96 * (i - 7) / 9 + 70 | i ≡ 7 mod 9 | 70, 166, 262, 358, ... |
7 | C[i,5] σ | 96 * (i - 7) / 27 + 22 | i ≡ 7 mod 27 | 22, 118, 214, 310, ... |
8 | C[i,6] σ | 192 * (i - 7) / 27 + 46 | i ≡ 7 mod 27 | 46, 238, 430, 622, ... |
9 | C[i,7] σ | 192 * (i - 61) / 81 + 142 | i ≡ 61 mod 81 | 142, 334, ... |
... | ... | ... | ... | ... |
j | C[i,j-2] σ | 6 * 2^{k+1} * (i - m) / 3^{l} + 3 * 2^{k} * h - 2 | i ≡ m mod 3^{l} | ... |
The general formula for a column j >= 4 uses the following parameters:
- k = floor(j / 2)
- l = floor(j - 1) / 2)
- m = a(floor((j - 1) / 4), where a(n) is the OEIS sequence (A066443: a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0 . The values are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides (4), 40, 364, 3280, 29524 (OEIS A191681). The constants appear first in columns 2-4 (in segment 1), 5-8 (in segment 7), 9-12 (in segment 61) and so on
- h = a(j), where a(n) is the OEIS sequence A084101 with period 4: a(0..3) = 1, 3, 3, 1; a(n) = a(n - 4) for n > 3.
(This results in k = 2, l = 1, m = 1, h = 1 for j = 4.)
The first few lines of the segment directory are the following:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ... | 2*j | 2*j+1 | |
i | 6*i‑2 | µµ | δµµ | µµσ | δµµσ | µµσσ | δµµσσ | µµσ^{3} | δµµσ^{3} | µµσ^{4} | δµµσ^{4} | ... | µµσ^{j-1} | δµµσ^{j-1} |
1 | 4 | 16 | 4 | 10 | ||||||||||
2 | 10 | 40 | ||||||||||||
3 | 16 | 64 | ||||||||||||
4 | 22 | 88 | 28 | 58 | ||||||||||
5 | 28 | 112 | ||||||||||||
6 | 34 | 136 | ||||||||||||
7 | 40 | 160 | 52 | 106 | 34 | 70 | 22 | 46 |
There is a more elaborated segment directory with 5000 rows.
Properties of the Segment Directory
We make a number of claims for the segment directory C:
- (C1) All nodes in the segment directory have the form 6 * n - 2.
- This follows from the formula for columns C[i,1..3], and for any higher column numbers from the 3-by-2 replacement property of the σ operation.
- (C2) All segments have a finite length.
- At some point the σ operations will have replaced all factors 3 by 2.
- (C3) All nodes in the right part of a segment have the form 6 * (3^{n} * 2^{m} * f) - 2 with the same "3-2-free" factor f.
- This follows from the operations for columns C[i,1..3], and from the fact that the σ operation maintains this property.
- (C4) All nodes in the right part of a particular segment are
- different among themselves, and
- different from the left side of that segment (except for the first segment for the root 4).
- For C[i,1..2] we see that the values modulo 24 are different. For the remaining columns, we see that the exponents of the factors 2 and 3 are different. They are shifted by the σ operations, but they alternate, for example (in the segment with left part 40):
160 = 6 * (3^{3} * 2^{0} * 1) - 2 52 = 6 * (3^{2} * 2^{0} * 1) - 2 106 = 6 * (3^{2} * 2^{1} * 1) - 2 34 = 6 * (3^{1} * 2^{1} * 1) - 2 70 = 6 * (3^{1} * 2^{2} * 1) - 2 22 = 6 * (3^{0} * 2^{2} * 1) - 2 46 = 6 * (3^{0} * 2^{3} * 1) - 2
- (C5) There is no cycle in a segment (except for the first segment for the root 4).
Segment Lengths
Oviously the segment directory is very structured. The lengths of the compressed segments follow the pattern
4 2 2 4 2 2 L_{1} 2 2 4 2 2 4 2 2 L_{2} 2 2 4 2 2 ...
with two fixed lengths 2 and 4 and some variable lengths L_{1}, L_{2} ... > 4. For the left parts 4, 40, 364, 3280, 29524 (OEIS A191681), the segment lengths have high values 4, 8, 12, 16, 20 which did not occur before. Those left parts are (9^{n+1} - 1) / 2, or 4 * Sum(9^{i}, i = 0..n).
Coverage of the Right Part
We now examine the modular conditions which result from the segment construction table (T2) in order to find out how the numbers of the form 6 * n - 2 are covered by the right part of the segment directory. The following table (T3) shows the result:
Columns j | Covered | Remaining |
---|---|---|
2-3 | 4, 16 mod 24 | 10, 22, 34, 46 mod 48 |
3-4 | 10, 34 mod 48 | 22, 46, 70, 94 mod 96 |
5-6 | 70, 22 mod 96 | 46, 94, 142, 190 mod 192 |
7-8 | 46, 142 mod 192 | 94, 190, 286, 382 mod 384 |
... | ... | ... |
We can always exclude the first and the third element remaining so far by looking in the next two columns of segments with sufficient length.
- (C6) There is no limit on the length of a segment.
- We only need to take a segment which, in its right part, has a factor of 3 with a sufficiently high power, and the σ operations will stretch out the segment accordingly.
Therefore we can continue the modulus table above indefinitely, which leads us to the claim:
- (C7) All numbers of the form 6 * n - 2 occur exactly once in the right part of the segment directory, and once as a left side. There is a bijective mapping between the left sides and the elements of the right parts.
- The sequences defined by the columns in the right part all have different modulus conditions. Therefore they are all disjoint. The left sides are disjoint by construction.
Segment Tree
So far we possess the segment directory C which represents the root segment and an infinite set of small trees with disjoint nodes and two branches. We know that the segments represent trees, and that their right parts are all disjoint and different from the left side.
We now want to attach (or connect) the segments to other graphs until we get a single big graph which will later become the backbone of the Collatz graph. Ideally the attachment process should maintain the tree property of the graphs all the time.
- The verb attach emphasizes the direction of the operation better than the verb connect.
Attachment Directory Construction
Parallel to the segment directory we maintain the attachment directory A which, for any source segment in C:
- tells whether the tree corresponding to the segment was already attached to the graph represented by some other segment, and if so,
- tells the target segment and column numbers in the segment directory C where the source segment was attached.
Initially all segments are unattached.
We operate on A as follows: Considering simultaneously a set of source segments i > 1 (i.e. omitting the root segment) in C - which fulfill some modularity condition (the source segment set), and which are so far unattached, we attach their segments parallel to the unique occurrences of their left sides in the right part of C (target segment set and target column).
Attachment rules
The following table (T4) tells the computation rules for the target position, depending on the modularity condition of the source segment. We identify and denote these attachment rules by the target column number. We show the the first segments (their left side) for k = 0, 1, 2, 3.
Rule / column |
Source segments |
Condition / remaining |
First source segments |
Target segments |
First target segments |
Dir. |
---|---|---|---|---|---|---|
5 | 6(2^{0}(4k + 3)) - 2 | 0 mod 8 2, 4, 6 mod 8 |
16, 40, 64, 88 | 6(3^{0}k + 1 ) - 2 | 4, 10, 16, 22 | < |
6 | 6(2^{0}(4k + 1)) - 2 | 4 mod 8 2, 6, 10, 14 mod 16 |
4, 28, 52, 76 | 6(3^{1}k + 1) - 2 | 4, 22, 40, 58 | < |
9 | 6(2^{1}(4k + 1)) - 2 | 10 mod 16 2, 6, 14 mod 16 |
10, 58, 106, 154 | 6(3^{1}k + 1) - 2 | 4, 22, 40, 58 | < |
10 | 6(2^{1}(4k + 3)) - 2 | 2 mod 16 6, 14, 22, 30 mod 32 |
34, 82, 130, 178 | 6(3^{2}k + 7) - 2 | 40, 94, 148, 202 | > |
13 | 6(2^{2}(4k + 3)) - 2 | 6 mod 32 14, 22, 30 mod 32 |
70, 166, 262, 358 | 6(3^{2}k + 7) - 2 | 40, 94, 148, 202 | < |
14 | 6(2^{2}(4k + 1)) - 2 | 22 mod 32 14, 30, 46, 62 mod 64 |
22, 118, 214, 310 | 6(3^{3}k + 7) - 2 | 40, 202, 364, 526 | > |
17 | 6(2^{3}(4k + 1)) - 2 | 46 mod 64 14, 30, 62 mod 64 |
46, 238, 430, 622 | 6(3^{3}k + 7) - 2 | 40, 202, 364, 526 | < |
18 | 6(2^{3}(4k + 3)) - 2 | 14 mod 64 30, 62, 94, 126 mod 128 |
142, 334, 526, 718 | 6(3^{4}k + 61) - 2 | 364, 850, 1336, 1822 | > |
21 | 6(2^{4}(4k + 3)) - 2 | 30 mod 128 62, 94, 126 mod 128 |
286, 670, 1054, 1438 | 6(3^{4}k + 61) - 2 | 364, 850, 1336, 1822 | > |
22 | 6(2^{4}(4k + 1)) - 2 | 94 mod 128 62, 126, 190, 254 mod 256 |
94, 478, 862, 1246 | 6(3^{5}k + 61) - 2 | 364, 1822, 3280, 4738 | > |
... | ... | ... | ... | ... | > | ... |
It should be obvious how the following rows of the table must be filled. The additive constants in the formula for the source segments follow the periodic pattern 3, 1, 1, 3 (OEIS A084101), while those for the target segments are taken from OEIS A066443. The latter constants change in every fourth row of (T4).
As an example, we apply rule 14 to source segment 22. (This example does not show the result of of the whole process, but only a single step.)
1 | 5 | 6 | 9 | 10 | 13 | 14 | 17 | 18 | 21 | 22 | ... | |
1 | 4 | 16 | 4 | 10 | ||||||||
2 | 10 | 40 | ||||||||||
3 | 16 | 64 | ||||||||||
4 | ||||||||||||
5 | 28 | 112 | ||||||||||
6 | 34 | 136 | ||||||||||
7 | 40 | 160 | 52 | 106 | 34 | 70 | 22 | 46 | ||||
22 | 88 | 28 | 58 |
Properties of the Attachment Rules
For the attachment directory A we note respectively claim:
- (A1) The source segments met by the conditions in the rules are all disjoint.
- (A2) Therefore, a source segment is chosen by the process exactly once.
- (A3) Each source segment meets a condition in some rule with a sufficiently high number.
- (A4) The construction is such that the target column always exists in the target segments.
- Table (T4) is derived from (T2) which has similiar modularity conditions.
- (A5) The target column (or rule number) depends on the modularity condition for the source segment alone, but not on (the left side of) the segment.
- This can be shown by the graph operations (δ / µ / σ) which are tied to the columns.
Shifting left or right
There are two categories of attachment rules (column Dir. in (T4)):
- Rules 5, 6, 9, 13, 17 attach to lower segments - they shift left.
- Rules 10, 14, 18 and above attach to higher segments - they shift right.
- This can be seen from the powers of 2 and 3 in the source and target row columns. Starting at segment 18, we have 3^{k} > 2^{k+2} for k >= 4.
With the single exception of the root segment 1, the rules obviously never attach a segment to itself.
Decreasing and increasing set of subtrees
Likewise, we can also group the subtrees which are built from the segments by attachment operations into two sets:
- the decreasing set T_{d} with members that will attach to some segment with a lower number, initially the segments for which rules 5, 6, 9, 13, 17 apply.
- the increasing set T_{i} with members that will attach to some segment with a higher number, initially the segments for which rules 10, 14, 18 and above apply.
We define that the root segment is also a member of T_{d}. The goal is the following claim: (A7) If T_{i} is empty, all segments in T_{d} finally attach to the tree above the root segment.
- Suppose n is the smallest member of T_{d} which is not yet attached. A left-shifting rule applies to n, but there is no smaller, unattached member in T_{d}. T_{i} is empty. Therefore n must be attached to the root or some segment in the tree above the root.
Reduction of the increasing set
We now try to move subsets of T_{i} to T_{d} by examining the parameter k in the formula for the targets t of some members s in T_{i} (c.f. (T4)). We concentrate on rule 10 because the targets of rules 14, 18 and above are a subset of the targets of rule 10 (i.e. the "longer" segments 4, 22, 40, 58, 76, 94 ...).
A simple observation is:
- (A8) We can move all members with even k.
- We attach s to t. For even k we get an odd factor t = 6*(2l + 1) - 2; t ≡ 0, 4 mod 8, so the left-shifting rules 5 or 6 apply to t, therefore t and the attached s are in T_{d}.
T_{i} now contains only (target) segments with odd k.
Rule sieving
Order of Rule Application
- (A7) The resulting graphs do not depend on the order of application of the attachment rules.
- The rules may well hit the same target segments, but they always do so in different target columns. It does not matter whether the target segment is already attached.
Despite of (A6) we will apply the rules in a well-defined order, because only in this order we can show that the tree property of the subgraphs is always maintained.
Attachment Process
We will now use the rules of (T4) to reduce the set of unattached segments in C in an iterative process. Our goal is to show that all segments are attached - mostly indirectly - to the root segment.
Supersegments
The segments considered so far contain nodes of the form 6 * i - 2. We call a node where i has the same form a supernode (of degree 2, 3, 4 and so on):
n_{1} = 6 * i - 2 = 6 * i - 2 ≡ 0 mod 2 n_{2} = 6 * (6 * i - 2) - 2 = 36 * i - 14 ≡ 2 mod 4: rules >= 9 n_{3} = 6 * (6 * (6 * i - 2) - 2) - 2 = 216 * i - 86 ≡ 2 mod 8; rules 9, 10 n_{4} = ... = 1296 * i - 518 ≡ 10 mod 16; rule 9 n_{5} = ... = 7776 * i - 3110 ≡ 10 mod 16; rule 9 ... n_{j} = 6^{j} * i - m_{j}
The additive constants m_{j} are taken here from OEIS A005610 with a(k) = 6 * a(k - 1) + 2 = 2 * (6 * 6^{k} - 1) / 5.
When a segment has a supernode as its left side, it is called a supersegment. An inspection of the segment directory C shows that supernodes occur at the following source positions (table (T5)):
Degree | Column | First source rows | Difference |
---|---|---|---|
2 | 1 | 4, 10, 16, 22 ... | 6 |
9 < | 4, 13, 22, 31 ... | 9 | |
10 > | 25, 52, 79, 106 ... | 27 | |
13 < | 16, 43, 70, 97 ... | 27 | |
... | ... | ... | |
3 | 1 | 22, 58, 94, 130 ... | 36 |
9 < | 22, 49, 76, 103 ... | 27 | |
10 > | 25, 106, 187, 268 ... | 81 | |
4 | 1 | 130, 346, 562, 778 ... | 216 |
9 < | 49, 130, 211, 292 ... | 81 | |
5 | 1 | 778, 2074, 3370 ... | 1286 |
9 < | 292, 778, 1264, 1750, 2236 ... | 486 = 6 * 81 | |
... | ... | ... | ... |
That are a rather simple consequences of the segment construction rules. We state some claims which are not so obvious:
- (S1) For degrees > 2, no other columns than the ones shown in table (T5) are occupied by supernodes of that degree.
- (S2) For degrees >= 4, only rule 9 (which moves downwards) is applicable.
- The property ≡ 10 mod 16 is maintained by the map i => 6 * i - 2 because 6 * 10 - 2 = 58 ≡ 10 mod 16.
- (S3) Supernodes only occur in segments s ≡ 4 mod 18. (These are the segments which have at least 6 columns).
- (S4) There is not more than one supernode in the right part of a segment, and if there is one, it occurs at the last or the last-but-one position in the right part (which represent the leafs of the corresponding trees).
- (S5) Each segment which contains a supernode in its right part:
- either has an odd row number,
- or a supernode as its left side.
- (S6) Each segment which does not contain a supernode in its right part (that are rows 1, 10, 19, 28, 37, 46, 55 ... i ≡ 1 mod 9):
- either has an odd row number,
- or a supernode as its left side.
We first attach all even rows mentioned in (S3). Then we attach the even rows mentioned in (S4).
No Cycles
- (A8) The attachment process does not create any new cycle (in addition to the one in the root segment).
- Let a segment/tree t_{1} with left side n_{1} and right part R_{1} be attached to node n_{1} in the right part R_{2} of the unique segment/tree t_{2} which has the left side by n_{2}. t_{1} and t_{2} are disjoint trees by (C4), therefore the result of such a single attachment step is a tree again (u_{2}, still with left side n_{2}).
Proof for the Collatz Tree
- (P1) The remaining single tree is a subgraph of the Collatz graph.
- The edges of the compressed tree carry combined operations µµ, δµµ and σ = δµ.
So far, numbers of the form x ≡ 0, 1, 2, 3, 5 mod 6 are missing from the compressed tree.
We insert intermediate nodes into the compressed tree by applying operations on the left parts of the segments as shown in the following table (T5):
Operation | Condition | Resulting Nodes | Remaining Nodes |
---|---|---|---|
δ | 2 * i - 1 | i ≡ 0, 2, 6, 8 mod 12 | |
µ | 12 * i - 4 | i ≡ 0, 2, 6 mod 12 | |
δµ | i ≡ 1, 2 mod 3 | 4 * i - 2 | i ≡ 0, 12 mod 24 |
δµµ | i ≡ 2 mod 3 | 8 * i - 4 | i ≡ 0 mod 24 |
δµµµ | i ≡ 2 mod 3 | 16 * i - 8 | (none) |
The first three rows in T5 care for the intermediate nodes at the beginning of the segment construction with columns 1, 2, 3. Rows 4 and 5 generate the sprouts (starting at multiples of 3) which are not contained in the segment directory.
We call such a construction a detailed segment (in contrast to the compressed segments described above).
- A detailed segment directory can be created by the same Perl program. In that directory, the two subpaths of a segment are shown in two lines. Only the highlighted nodes are unique.
- (P2) The connectivity of the compressed tree remains unaffected by the insertions.
- (P3) With the insertions of (T5), the compressed tree covers the whole Collatz graph.
- (P4) The Collatz graph is a tree (except for the cycle 4-2-1.