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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.
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| 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.
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|
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| 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
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| * always to m * 2, and
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| * to (m - 1) / 3 if m ≡ 1 mod 3.
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| When m ≡ 0 mod 3, the path will continue with duplications only, since these maintain the divisibility by 3.
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| | |
| 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 be concerned with this cycle, and we start with node 4, the ''root''.
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| ===Graph Operations===
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| 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):
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| {| class="wikitable" style="text-align:center"
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| !Name !! Mnemonic !! Distance to root !! Mapping !! Condition
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| |-
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| | d || down || -1 || m ↦ m / 2 || m ≡ 0 mod 2
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| |-
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| | u || up || -1 || m ↦ 3 * m + 1 || (none)
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| |-
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| | s := ud || spike || -2 || m ↦ (m / 2) * 3 + 1 || m ≡ 0 mod 2
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| |-
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| | δ || divide || +1 || m ↦ (m - 1) / 3 || m ≡ 1 mod 3
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| |-
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| | µ || multiply || +1 || m ↦ m * 2 || (none)
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| |-
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| | σ := δµ|| squeeze || +2 || m ↦ ((m - 1) / 3) * 2 || m ≡ 1 mod 3
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| |}
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| 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===
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| The σ operation, applied to numbers of the form 6 * m - 2, has an interesting property:
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| (6 * (3 * n) - 2) σ = 4 * 3 * n - 2 = 6 * (2 * n) - 2
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| 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. In the opposite direction, the s operation replaces a factor 2 in m by 3. It "squeezes" a 3 into a 2.
| |
| ===Kernels===
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| 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.
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| * The kernel is not affected by σ and s operations.
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| ==Segment Construction==
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| We will now construct 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 is 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''. One row i of C is constructed according to the following table (T2):
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| {| class="wikitable" style="text-align:left"
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| !Column !! Operation !! Formula !! Condition !! Sequence
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| |-
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| | <nowiki>C[i,1]</nowiki> || || 6 * i - 2 || || 4, 10, 16, 22, 28, ...
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| |-
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| | <nowiki>C[i,2]</nowiki> || <nowiki>C[i,1]</nowiki> µµ || 24 * (i - 1) + 16 || || 16, 40, 64, 88, 112, ...
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| |-
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| | <nowiki>C[i,3]</nowiki> || <nowiki>C[i,1]</nowiki> δµµ || 24 * (i - 1) / 3 + 4 || i ≡ 1 mod 3 || 4, 28, 52, 76, 100, ...
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| |-
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| | <nowiki>C[i,4]</nowiki> || <nowiki>C[i,2]</nowiki> σ || 48 * (i - 1) / 3 + 10 || i ≡ 1 mod 3 || 10, 58, 106, 134, ...
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| |-
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| | <nowiki>C[i,5]</nowiki> || <nowiki>C[i,3]</nowiki> σ || 48 * (i - 7) / 9 + 34 || i ≡ 7 mod 9 || 34, 82, 130, 178, ...
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| |-
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| | <nowiki>C[i,6]</nowiki> || <nowiki>C[i,2]</nowiki> σσ || 96 * (i - 7) / 9 + 70 || i ≡ 7 mod 9 || 70, 166, 262, 358, ...
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| |-
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| | <nowiki>C[i,7]</nowiki> || <nowiki>C[i,3]</nowiki> σσ || 96 * (i - 7) / 27 + 22 || i ≡ 7 mod 27 || 22, 118, 214, 310, ...
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| |-
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| | <nowiki>C[i,8]</nowiki> || <nowiki>C[i,2]</nowiki> σσσ || 192 * (i - 7) / 27 + 46 || i ≡ 7 mod 27 || 46, 238, 430, 622, ...
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| |-
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| | <nowiki>C[i,9]</nowiki> || <nowiki>C[i,3]</nowiki> σσσ || 192 * (i - 61) / 81 + 142|| i ≡ 61 mod 81 || 142, 334, ...
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| |-
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| | ... || ... || ... || ... || ...
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| |-
| |
| |}
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| 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''. We make a number of claims for such segments:
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| * (C1) All nodes in the segment directory are of the form 6 * n - 2.
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| : 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.
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| * (C2) All segments have a finite length.
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| : At some point the σ operations will have replaced all factors 3 by 2.
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| * (C3) All nodes in the right part of a segment have the same kernel. | |
| : This follows from the formula for columns <nowiki>C[i,1..3]</nowiki>, and from the fact that the σ operation maintains this property.
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| * (C4) All nodes in the right part of a segment are different.
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| : 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 (right part of segment starting with 40, kernel 1):
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| 160 = 6 * 3<sup>3</sup> * 2<sup>0</sup> - 2
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| 52 = 6 * 3<sup>2</sup> * 2<sup>0</sup> - 2
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| 106 = 6 * 3<sup>2</sup> * 2<sup>1</sup> - 2
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| 34 = 6 * 3<sup>1</sup> * 2<sup>1</sup> - 2
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| 70 = 6 * 3<sup>1</sup> * 2<sup>2</sup> - 2
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| 22 = 6 * 3<sup>0</sup> * 2<sup>2</sup> - 2
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| 46 = 6 * 3<sup>0</sup> * 2<sup>3</sup> - 2
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| * (C5) There is no cycle in a segment.
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| ===Segment Lengths===
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| The segment directory is obviously very structured. The lengths of the compressed segments follow the pattern
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| 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 ...
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| 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).
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| ===Coverage of the Right Part===
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| 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 floowing table (T3):
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| {| class="wikitable" style="text-align:left"
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| !Columns j !! Covered !! Remaining
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| |-
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| | 2-3 || 4, 16 mod 24 || 10, 22, 34, 46 mod 48
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| |-
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| | 3-4 || 10, 34 mod 48 || 22, 46, 70, 94 mod 96
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| |-
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| | 5-6 || 70, 22 mod 96 || 46, 94, 142, 190 mod 192
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| |-
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| | 7-8 || 46, 142 mod 192 || 94, 190, 286, 382 mod 384
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| |-
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| | ... || ... || ...
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| |}
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| 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.
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| * (C6) There is no limit on the length of a segment.
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| : 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.
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| Therefore we can continue the modulus table above indefinitely, which leads us to the claim:
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| * (C7) '''All numbers of the form 6 * n - 2 occur exactly once''' in the right part <nowiki>C[i,j], j > 1</nowiki> of the segment directory.
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| : The sequences defined by the columns in the right part all have differen modulus conditions. Therefore they are all disjoint.
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| ===Segment Connectivity Sieve===
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| The segments are small trees with two branches. We now do 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 in C fulfilling 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).
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| : 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.
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| We ''sieve'' the segments: Once we have attached a set, we strike it out of the list still to be examined. The attachment rules are shown in the following table (T4):
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| {| class="wikitable" style="text-align:left"
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| |-
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| !Source Row i !! Target Row !! Column !! Remaining Rows !! Fraction
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| |-
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| |i ≡ 3 mod 4 ||((i - 3) / 4) * 3<sup>0</sup> + 1 || 2||i ≡ 0, 1, 2 mod 4 ||3/4
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| |-
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| |i ≡ 1 mod 4 ||((i - 1) / 4) * 3<sup>1</sup> + 1 || 3||i ≡ 0, 2, 4, 6 mod 8 ||1/2
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| |-
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| |i ≡ 2 mod 8 ||((i - 2) / 8) * 3<sup>1</sup> + 1 || 4||i ≡ 0, 4, 6 mod 8 ||3/8
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| |-
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| |i ≡ 6 mod 8 ||((i - 6) / 8) * 3<sup>2</sup> + 7 || 5||i ≡ 0, 4, 8, 12 mod 16 ||1/4
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| |-
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| |i ≡ 12 mod 16||((i - 12) / 16) * 3<sup>2</sup> + 7 || 6||i ≡ 0, 4, 8 mod 16 ||3/16
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| |-
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| |i ≡ 4 mod 16||((i - 4) / 16) * 3<sup>3</sup> + 7 || 7||i ≡ 0, 8, 16, 24 mod 32 ||1/8
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| |-
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| |i ≡ 8 mod 32||((i - 8) / 32) * 3<sup>3</sup> + 7 || 8||i ≡ 0, 16, 24 mod 32 ||3/32
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| |-
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| |i ≡ 24 mod 32||((i - 24) / 32) * 3<sup>4</sup> + 61|| 9||i ≡ 0, 16, 32, 48 mod 64||1/16
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| |-
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| |i ≡ 48 mod 64||((i - 48) / 64) * 3<sup>4</sup> + 61||10||i ≡ 0, 16, 32 mod 64 ||3/64
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| |-
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| | ... || ... || ... || ... || ...
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| |-
| |
| |}
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| The ellipsis row of this table could be explained in terms of i, but it should be obvious how must be constructed. The residues of 2<sup>k</sup> in the first column are 2<sup>k-2</sup> * 3, 2<sup>k-2</sup> * 1 in alternating sequence. The additive constants in the second column are the indexes of the variable 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 "jumps" by 4.
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| ===Segment Tree===
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| * (C8) The construction does not create any cycle.
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| : A cycle would contain a node which is entered by two edges. That would violate the uniqueness of the nodes in the right part of the segment directory.
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| * (C9) Any source row can be attached to the root segment.
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| : As for the 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).
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| * (C10) In the end, the graph of attached segments would be a one tree starting at 4. | |
| <!--
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| Example: Segment for 4 with segments for 16 and 10 attached:
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| 4 -> 16 -> 4 -> 10
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| | |
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| V V
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| 64 -> 256 40 -> 160 -> 52 -> 106 -> 34 -> 70 -> 22 -> 46
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| ...
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| -->
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| ===Detailed Segment Directory===
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| Once the coverage, connectivity and cycle-freeness of the segment tree is established, we switch back to the full Collatz graph. We feaze the segment, and we list the intermediate results of the operations in two separate rows:
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| {| class="wikitable" style="text-align:center"
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| |-
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| |6*i-2 || µ || µµ || µµδ || µµδµ || µµδµδ || µµδµδµ || ...
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| |-
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| | || δ || δµ || δµµ || δµµδ || δµµδµ || δµµδµδ || ...
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| |}
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| 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 easily be created by a [https://github.com/gfis/fasces/blob/master/oeis/collatz/collatz_rails.pl Perl program].
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| * The connections between the segments via the nodes 6 * i - 2 remain unaffected.
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| From a segment start of the form 6 * i - 2 we reach the numbers x according to the following table (T5):
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| {| class="wikitable" style="text-align:left"
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| |-
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| ! Number !! Operation !! Remark
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| |-
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| | x ≡ 1, 3, 5 mod 6 || (6 * i - 2) δ = 2 * i - 1 || all odd numbers
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| |-
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| | x ≡ 0 mod 6 || (3<sup>p</sup>) µ<sup>q</sup> || odd multiples of 3 times powers of 2
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| |-
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| | x ≡ 8 mod 12 || (6 * i - 2) µ || 8, 20, 32, 44, ...
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| |-
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| | x ≡ 2 mod 12 || (6 * i - 2) δµ if i ≡ 1 mod 3 || 2, 14, 26, 38, ...
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| |-
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| |}
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| Together this shows that we can reach all numbers x ≡ 0, 1, 2, 3, 5 mod 6 which are not contained in the right part of the compressed segment directory.
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| ===Duplicated Nodes===
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| This straightforward construction of the detailed segments introduces duplications of nodes in the right part. We make the following observations:
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| * Odd numbers occur exactly twice, once in column <nowiki>S[i,2]</nowiki>, and once in the right part.
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| * Iff a number ≡ 2 mod 6 occurs at the end of an S row, it occurs twice.
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