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| ==Introduction==
| | A collection of considerations regarding the '''[https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture]'''. |
| Collatz sequences are sequences of non-negative integer numbers with a simple construction rule:
| |
| :Even elements are halved, and odd elements 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 any start value. This problem is the '''Collatz conjecture''', for which the [https://en.wikipedia.org/wiki/Collatz_conjecture 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.
| |
|
| |
|
| Straightforward visualizations of the Collatz sequences no obvious structure. The sequences for the first dozen of start values are rather short, but the sequence for 27 suddenly has 112 elements.
| | It is splitted into the following parts: |
| ===References===
| | * [[OEIS/3x%2B1_Intro]] |
| * Jeffry C. Lagarias, Ed.: ''The Ultimate Challenge: The 3x+1 Problem'', Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. [http://www.ams.org/bookpages/mbk-78 MBK78]
| | * [[OEIS/3x%2B1_Segments]] |
| * OEIS A07165: [http://oeis.org/A070165/a070165.txt File of first 10K Collatz sequences], ascending start values, with lengths
| | * [[OEIS/3x%2B1_Connectivity]] |
| * Manfred Trümper: ''The Collatz Problem in the Light of an Infinite Free Semigroup''. Chinese Journal of Mathematics, Vol. 2014, [http://dx.doi.org/10.1155/2014/756917 Article ID 756917], 21 p. | | * [[OEIS/3x%2B1_Levels]] |
| ==Collatz Graph==
| | * [[OEIS/Attaching segments]] |
| 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 trivial cycle, and we start the graph with node 4, the ''root''.
| |
| Moreover, a trivial path starts when m ≡ 0 mod 3. We call such a path a ''sprout'', and it contains duplications only. A sprout must be added to the graph for any node divisible by 3, therefore we will not consider them for the moment.
| |
| ===Graph Operations===
| |
| Following [http://dx.doi.org/10.1155/2014/756917 Trümper], we use abbreviations for the elementary operations which transform a node (element, number) in the Collatz graph according to the following table (T1):
| |
| {| class="wikitable" style="text-align:center"
| |
| !Name !! Mnemonic !! Distance to root !! Mapping !! Condition
| |
| |-
| |
| | d || down || -1 || m ↦ m / 2 || m ≡ 0 mod 2
| |
| |-
| |
| | u || up || -1 || m ↦ 3 * m + 1 || (none)
| |
| |-
| |
| | s := ud || spike || -2 || m ↦ (m / 2) * 3 + 1 || m ≡ 0 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.
| |
| <!--
| |
| === Trivial paths===
| |
| There are two types of paths whose descriptions are very simple:
| |
| (n = 2<sup>k</sup>) ddd ... d 8 d 4 d 2 d 1 - powers of 2
| |
| (n ≡ 0 mod 3) uuu ... u (n * 2<sup>k</sup>) ... - multiples of 3
| |
| ===Kernels===
| |
| By the ''kernel'' of a number n = 6 * m - 2 we denote the "2-3-free" factor of m, that is the factor which remains when all powers of 2 and 3 have been removed from m.
| |
| * The kernel is not affected by σ and s operations.
| |
| -->
| |
| ===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 [https://oeis.org/A070165 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.
| |
| Walking 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.
| |
| | |
| ==Segment Construction==
| |
| 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. We assemble the segments as rows of an infinite array <nowiki>C[i,j]</nowiki>, 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 "detailled" 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.
| |
| The columns in one row i of the array C are constructed as described in the following table (T2):
| |
| {| class="wikitable" style="text-align:left"
| |
| !Column j !! Operation !! Formula !! Condition !! Sequence
| |
| |-
| |
| | 1 || || 6 * i - 2 || || 4, 10, 16, 22, 28, ...
| |
| |-
| |
| | 2 || <nowiki>C[i,1]</nowiki> µµ || 24 * (i - 1) + 16 || || 16, 40, 64, 88, 112, ...
| |
| |-
| |
| | 3 || <nowiki>C[i,1]</nowiki> δµµ || 24 * (i - 1) / 3 + 4 || i ≡ 1 mod 3 || 4, 28, 52, 76, 100, ...
| |
| |-
| |
| | 4 || <nowiki>C[i,2]</nowiki> σ || 48 * (i - 1) / 3 + 10 || i ≡ 1 mod 3 || 10, 58, 106, 134, ...
| |
| |-
| |
| | 5 || <nowiki>C[i,3]</nowiki> σ || 48 * (i - 7) / 9 + 34 || i ≡ 7 mod 9 || 34, 82, 130, 178, ...
| |
| |-
| |
| | 6 || <nowiki>C[i,2]</nowiki> σσ || 96 * (i - 7) / 9 + 70 || i ≡ 7 mod 9 || 70, 166, 262, 358, ...
| |
| |-
| |
| | 7 || <nowiki>C[i,3]</nowiki> σσ || 96 * (i - 7) / 27 + 22 || i ≡ 7 mod 27 || 22, 118, 214, 310, ...
| |
| |-
| |
| | 8 || <nowiki>C[i,2]</nowiki> σσσ || 192 * (i - 7) / 27 + 46 || i ≡ 7 mod 27 || 46, 238, 430, 622, ...
| |
| |-
| |
| | 9 || <nowiki>C[i,3]</nowiki> σσσ || 192 * (i - 61) / 81 + 142|| i ≡ 61 mod 81 || 142, 334, ...
| |
| |-
| |
| | ... || ... || ... || ... || ...
| |
| |-
| |
| |}
| |
| The first column(s) <nowiki>C[i,1]</nowiki> will be denoted as the ''left part'' of the segment (or of the whole directory), while the columns <nowiki>C[i,j], j > 1</nowiki> will be the ''right part''. The first few lines of the segment directory are the following:
| |
| | |
| <table style="border-collapse: collapse; ">
| |
| <tr>
| |
| <td style="text-align:center"> </td>
| |
| <td style="text-align:center">1</td>
| |
| <td style="text-align:center">2</td>
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| <td style="text-align:center">3</td>
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| <td style="text-align:center">4</td>
| |
| <td style="text-align:center">5</td>
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| <td style="text-align:center">6</td>
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| <td style="text-align:center">7</td>
| |
| <td style="text-align:center">8</td>
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| <td style="text-align:center">9</td>
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| <td style="text-align:center">10</td>
| |
| <td style="text-align:center">11</td>
| |
| <td style="text-align:center">...</td>
| |
| <td style="text-align:center">2*j</td>
| |
| <td style="text-align:center">2*j+1</td>
| |
| </tr>
| |
| <tr>
| |
| <td style="border:1px solid gray;text-align:right" > i </td>
| |
| <td style="border:1px solid gray;text-align:right" >6*i‑2</td>
| |
| <td style="border:1px solid gray;text-align:right" >µµ</td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµ</td>
| |
| <td style="border:1px solid gray;text-align:right" >µµσ</td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµσ</td>
| |
| <td style="border:1px solid gray;text-align:right" >µµσσ</td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµσσ</td>
| |
| <td style="border:1px solid gray;text-align:right" >µµσ<sup>3</sup></td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµσ<sup>3</sup></td>
| |
| <td style="border:1px solid gray;text-align:right" >µµσ<sup>4</sup></td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµσ<sup>4</sup></td>
| |
| <td style="border:1px solid gray;text-align:right" >...</td>
| |
| <td style="border:1px solid gray;text-align:right" >µµσ<sup>j-1</sup></td>
| |
| <td style="border:1px solid gray;text-align:right" >δµµσ<sup>j-1</sup></td>
| |
| </tr>
| |
| <tr><td> 1 </td><td style="border:1px solid gray;text-align:right" > 4 </td><td style="border:1px solid gray;text-align:right" > 16 </td><td style="border:1px solid gray;text-align:right" title="1.0:0" id="4" class="d4 bor seg"> 4 </td><td style="border:1px solid gray;text-align:right" title="1.1:0" id="10" class="d4 bor seg"> 10 </td></tr>
| |
| <tr><td> 2 </td><td style="border:1px solid gray;text-align:right" > 10 </td><td style="border:1px solid gray;text-align:right" > 40 </td></tr>
| |
| <tr><td> 3 </td><td style="border:1px solid gray;text-align:right" > 16 </td><td style="border:1px solid gray;text-align:right" > 64 </td></tr>
| |
| <tr><td> 4 </td><td style="border:1px solid gray;text-align:right" > 22 </td><td style="border:1px solid gray;text-align:right" > 88 </td><td style="border:1px solid gray;text-align:right" title="5.0:0" id="28" class="d4 bor seg"> 28 </td><td style="border:1px solid gray;text-align:right" title="5.1:0" id="58" class="d4 bor seg"> 58 </td></tr>
| |
| <tr><td> 5 </td><td style="border:1px solid gray;text-align:right" > 28 </td><td style="border:1px solid gray;text-align:right" > 112 </td></tr>
| |
| <tr><td> 6 </td><td style="border:1px solid gray;text-align:right" > 34 </td><td style="border:1px solid gray;text-align:right" > 136 </td></tr>
| |
| <tr><td> 7 </td><td style="border:1px solid gray;text-align:right" > 40 </td><td style="border:1px solid gray;text-align:right" > 160 </td><td style="border:1px solid gray;text-align:right" title="1.0:2" id="52" class="d4 bor seg"> 52 </td><td style="border:1px solid gray;text-align:right" title="1.1:2" id="106" class="d4 bor seg"> 106 </td><td style="border:1px solid gray;text-align:right" title="1.1:1" id="34" class="d4 bor seg"> 34 </td><td style="border:1px solid gray;text-align:right" title="1.2:1" id="70" class="d4 bor seg"> 70 </td><td style="border:1px solid gray;text-align:right" title="1.2:0" id="22" class="d4 bor seg"> 22 </td><td style="border:1px solid gray;text-align:right" title="1.3:0" id="46" class="d4 bor seg"> 46 </td></tr>
| |
| </table>
| |
| A more elaborated '''[http://www.teherba.org/fasces/oeis/collatz/comp.html segment directory with 30000 rows]''' can easily be generated by this [https://github.com/gfis/fasces/blob/master/oeis/collatz/collatz_rails.pl Perl program].
| |
| ===Properties of the segment directory===
| |
| We make a number of claims for segments:
| |
| * (C1) All nodes in the segment directory are of the form 6 * n - 2. | |
| : This follows from the formula for columns <nowiki>C[i,1..3]</nowiki>, 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<sup>n</sup> * 2<sup>m</sup> * f) - 2 with the same "3-2-free" factor f.
| |
| : This follows from the operations for columns <nowiki>C[i,1..3]</nowiki>, and from the fact that the σ operation maintains this property.
| |
| * (C4) All nodes in the right part of a segment are different.
| |
| : For <nowiki>C[i,1..2]</nowiki> 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<sup>3</sup> * 2<sup>0</sup> * 1) - 2
| |
| 52 = 6 * (3<sup>2</sup> * 2<sup>0</sup> * 1) - 2
| |
| 106 = 6 * (3<sup>2</sup> * 2<sup>1</sup> * 1) - 2
| |
| 34 = 6 * (3<sup>1</sup> * 2<sup>1</sup> * 1) - 2
| |
| 70 = 6 * (3<sup>1</sup> * 2<sup>2</sup> * 1) - 2
| |
| 22 = 6 * (3<sup>0</sup> * 2<sup>2</sup> * 1) - 2
| |
| 46 = 6 * (3<sup>0</sup> * 2<sup>3</sup> * 1) - 2
| |
| * (C5) There is no cycle in a segment.
| |
| ===Segment Lengths===
| |
| The segment directory is obviously very structured. The lengths of the compressed segments follow the pattern
| |
| 4 2 2 4 2 2 L<sub>1</sub> 2 2 4 2 2 4 2 2 L<sub>2</sub> 2 2 4 2 2 ...
| |
| with two ''fixed lengths'' 2 and 4 and some ''variable lengths'' L<sub>1</sub>, L<sub>2</sub> ... > 4. For the left parts 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]), the segment lengths have high values 4, 8, 12, 16, 20 which did not occur before. Those left parts are (9<sup>n+1</sup> - 1) / 2, or 4 * Sum(9<sup>i</sup>, i = 0..n).
| |
| ===Coverage of the Right Part===
| |
| We now examine the modular conditions which result from the segment construction table in order to find out how the numbers of the form 6 * n - 2 are covered by the right part of the segment directory, as shown in the following table (T3):
| |
| {| class="wikitable" style="text-align:left"
| |
| !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.
| |
| : The sequences defined by the columns in the right part all have different modulus conditions. Therefore they are all disjoint.
| |
| ==Segment Connectivity Sieve==
| |
| The segments represent small trees with two branches. We now start an ''Gedankenexperiment'' analogous to [https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel Hilbert's hotel]. We consider simultaneously all rows i > 1 (omitting the root) in C which fulfill some modularity condition (the ''source'' row in C), and we ''attach'' (identify, connect) them to their unique occurrence in the right part of C (''target'' row and column).
| |
| : A technical implementation of the process (which is impossible for infinite sets) would perhaps replace the target node by a pointer to the source segment.
| |
| We ''sieve'' the segments: Once we have attached a set of segments, we strike it out of the list still to be examined. The attachment rules are shown in the following table (T4):
| |
| {| class="wikitable" style="text-align:left"
| |
| |-
| |
| !Source Row i !! Target Row !! Column !! Remaining Rows !! Fraction
| |
| |-
| |
| |i ≡ 3 mod 4 ||((i - 3) / 4) * 3<sup>0</sup> + 1 || 2||i ≡ 0, 1, 2 mod 4 ||3/4
| |
| |-
| |
| |i ≡ 1 mod 4 ||((i - 1) / 4) * 3<sup>1</sup> + 1 || 3||i ≡ 0, 2, 4, 6 mod 8 ||1/2
| |
| |-
| |
| |i ≡ 2 mod 8 ||((i - 2) / 8) * 3<sup>1</sup> + 1 || 4||i ≡ 0, 4, 6 mod 8 ||3/8
| |
| |-
| |
| |i ≡ 6 mod 8 ||((i - 6) / 8) * 3<sup>2</sup> + 7 || 5||i ≡ 0, 4, 8, 12 mod 16 ||1/4
| |
| |-
| |
| |i ≡ 12 mod 16||((i - 12) / 16) * 3<sup>2</sup> + 7 || 6||i ≡ 0, 4, 8 mod 16 ||3/16
| |
| |-
| |
| |i ≡ 4 mod 16||((i - 4) / 16) * 3<sup>3</sup> + 7 || 7||i ≡ 0, 8, 16, 24 mod 32 ||1/8
| |
| |-
| |
| |i ≡ 8 mod 32||((i - 8) / 32) * 3<sup>3</sup> + 7 || 8||i ≡ 0, 16, 24 mod 32 ||3/32
| |
| |-
| |
| |i ≡ 24 mod 32||((i - 24) / 32) * 3<sup>4</sup> + 61|| 9||i ≡ 0, 16, 32, 48 mod 64||1/16
| |
| |-
| |
| |i ≡ 48 mod 64||((i - 48) / 64) * 3<sup>4</sup> + 61||10||i ≡ 0, 16, 32 mod 64 ||3/64
| |
| |-
| |
| |i ≡ 16 mod 64||((i - 16) / 64) * 3<sup>5</sup> + 61||11||i ≡ 0, 32, 64, 96 mod 128 ||1/32
| |
| |-
| |
| | ... || ... || ... || ... || ...
| |
| |-
| |
| |}
| |
| The ellipsis row of this table could be explained in terms of i, but it should be obvious how it can be constructed. The residues of 2<sup>k</sup> in the first column are 3 * 2<sup>k-2</sup>, 1 * 2<sup>k-2</sup> in an alternating sequence. The additive constants in the second column are the indexes of the variable length segments with left parts (4), 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]) mentioned above. They are repeated 4 times since the corresponding lengths "jump" by 4.
| |
| ====Example====
| |
| 82 is a node with a long Collatz sequence. The following list (L1) shows - via the highlighted numbers - how segments are attached:
| |
| 1 2 3 4 5 6 7 8 9 10 11 12
| |
| C[ 14] '''82''' 328
| |
| C[ 16] '''94''' 376 124 250 '''82''' 166
| |
| C[ 61] '''364''' 1456 484 970 322 646 214 430 142 286 '''94''' 190
| |
| C[ 46] '''274''' 1096 '''364''' 730
| |
| C[ 52] '''310''' 1240 412 826 '''274''' 550
| |
| C[ 88] '''526''' 2104 700 1402 466 934 '''310''' 622
| |
| C[223] '''1336''' 5344 1780 3562 1186 2374 790 1582 '''526''' 1054
| |
| C[ 56] '''334''' '''1336 '''
| |
| C[142] '''850''' 3400 1132 2266 754 1510 502 1006 '''334''' 670
| |
| C[160] '''958''' 3832 1276 2554 '''850''' 1702
| |
| C[304] '''1822''' 7288 2428 4858 1618 3238 1078 2158 718 1438 478 '''958 '''
| |
| C[385] '''2308''' 9232 3076 6154 2050 4102 1366 2734 910 '''1822 '''
| |
| C[289] '''1732''' 6928 '''2308''' 4618
| |
| C[217] '''1300''' 5200 '''1732''' 3466
| |
| C[163] '''976''' 3904 '''1300''' 2602
| |
| C[ 41] '''244''' '''976 '''
| |
| C[ 31] '''184''' 736 '''244''' 490
| |
| C[ 8] '''46''' '''184 '''
| |
| C[ 7] '''40''' 160 52 106 34 70 22 '''46 '''
| |
| C[ 2] '''10''' '''40 '''
| |
| C[ 1] '''4''' 16 4 '''10 '''
| |
| ===Segment Tree===
| |
| * (C8) The process does not create any cycle.
| |
| : All involved subtrees are disjoint. The process attaches trees to other trees, the results are trees again.
| |
| * (C9) Any source row will finally be attached to the tree starting at the root segment.
| |
| : Similiar to the arguments for coverage, we have to apply the rules from table T4 one after the other up to a sufficiently high row (corresponding to a sufficiently long variable segment).
| |
| | |
| ==Collatz Tree==
| |
| * (C10) The segment tree is a subgraph of the Collatz graph.
| |
| : The edges of the segment 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 segment tree.
| |
| | |
| We insert intermediate nodes into the segment tree by applying operations on the left parts of the segments as shown in the following table (T5):
| |
| {| class="wikitable" style="text-align:left"
| |
| |-
| |
| ! 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 '''[http://www.teherba.org/fasces/oeis/collatz/rails.html detailed segment directory]''' can be created by the same [https://github.com/gfis/fasces/blob/master/oeis/collatz/collatz_rails.pl Perl program]. In that directory, the two subpaths of a segment are shown in two lines. Only the highlighted nodes are unique.
| |
| | |
| * (C11) The connectivity of the segment tree remains unaffected by the insertions.
| |
| * (C12) With the insertions of T5, the segment tree covers the whole Collatz graph.
| |
| * (C13) '''The Collatz graph is a tree''' (except for the trivial cycle).
| |