Collatz Streetmap: Difference between revisions

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imported>Gfis
new def. of road
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* such that the reader could finally be convinced that the algorithm enumerates all numbers.
* such that the reader could finally be convinced that the algorithm enumerates all numbers.
===Steps===
===Steps===
 
In order to walk back and forth in the Collatz graph, we need to describe how to move from one node in the graph to another. We will write
a ''s'' b
when use step ''s'' to move from node (element) a in the Collatz graph/sequence to node b. The following table lists the steps which we will use:
{| class="wikitable" style="text-align:center"
!Name !! Mnemonic !! Direction !! Operation        !! Condition          !! Remark
|-
| h  || halve    ||  -> root  ||  b = a / 2      || a ≡ 0 mod 2  || next in sequence
|-
| t  || triple  ||  -> root  ||  b = 3 * a + 1  || true                || next in sequence
|-
| m  || multiply ||  root ->  ||  b = a * 2      || true                || next up in tree
|-
| d  || divide  ||  root ->  ||  b = (a - 1) / 3 || a ≡ 1 mod 3  || next up in tree
|}
==Collatz roads==
===Motivation: Patterns in sequences with same length===
===Motivation: Patterns in sequences with same length===
When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, and with a characteristical neighbourhood of every other value, for example (from [https://oeis.org/A070165 OEIS A070165]):
When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, for example (from [https://oeis.org/A070165 OEIS A070165]):
  142/104: [142 m 71 d 214 m 107 d 322 m 161 d 484 m 242 m 121 d | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
  142/104: [142 h 71 t 214 h 107 t 322 h 161 t 484 h 242 h 121 t 364 | 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
143/104: [143 t 430 h 215 t 646 h 323 t 970 h 485 t 1456 h 728 h 364 | 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
             +1  *6+4    +1  *6+4    +1  *6+4    +1  *6+4  *6+2      =    =  ...
             +1  *6+4    +1  *6+4    +1  *6+4    +1  *6+4  *6+2      =    =  ...
143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
The third line shows the relation between the values in the first two lines.  
==Collatz roads==
 
We define a "road" (with 2 parallel "lanes") as a sequence of pairs of elements (in 2 Collatz sequences with adjacent start values, read from right to left). A road is constructed by taking some n (364 in the example, the last common element of the 2 sequences) with n ≡ 4 mod 6, and by applying the steps
Informally, a '''road''' is a parallel arrangement of 2 pathes stemming from 2 sequences which have a common tail. In the example above, the road starts with an element ≡ 4 mod 6 (the 364 before the bar), followed by t/h and h/h steps, and the road then extends to the left as long as a characteristical, alternating sequence of pairs of steps ht - th - ht - th ... can be continued. In the example, the road can be continued to the left with 4 additional steps, but the construction then stops since there is no number q such that q * 3 + 1 = 62:
d m m d m d m d ...  
  q t | 62 h 31 t  94 h  47 t 142 h ...
m m d m d m d m ...
126 h | 63 t 190 h 95 t 286 h 143 t ...
in alternating sequence, until one of the elements in the pairs becomes divisible by 3. The construction yields one road with an upper lane (left elements of the pairs) and a lower lane (right elements), for example:  
        +1  *6+4    +1  *6+4    +1 
  ''364'': (121,728), (242,''1456''), (''484'',485), (161,''970''), (''322'',323), (107,''646''), (''214'',215), (71,''430''), (''142'',143) ...
The elements which are ≡ 4 mod 6 are emphasized. The construction continues until 63 ≡ 0 mod 3:
  ... (47,''286''), (''94'',95), (31,''190''), (62,63).
===Roads table R===
===Roads table R===
For better handling (e.g. in Excel), the roads for all starting values 4, 10, 16, 22 ... n*6+4 are listed as rows of a table. The columns in the rows are numbered r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub> ...
For easier handling (e.g. in Excel) we list the roads for all starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are numbered r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub> ...
<table>
<table>
<tr align="right">
<tr align="right">
Line 113: Line 125:
<tr align="right"><td><strong>76</strong></td><td class="arr">4</td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>
<tr align="right"><td><strong>76</strong></td><td class="arr">4</td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>
</table>
</table>
There is a more '''[http://www.teherba.org/fasces/oeis/collatz/roads.html elaborated example]''' for elements &lt;= 143248. The zoom factor of the web browser may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of roads can be seen.
The roads contain 2 parallel, short, interrelated paths in the Collatz graph. Though the graph usually has a "chaotic" appearance, the roads exhibit quite some amount of regular structure.
===Relation to the Collatz graph===
The Collatz graph is the union of all possible Collatz sequences, read backwards. Apart from the initial cycle 1-2-4-1 it should be a tree, and it should contain all positive numbers.  


Each road contains 2 short paths in the Collatz graph, the root is left of the starting value of the road, and both proceed "upwards" in the tree. In contrast to the usual "chaotic" appearance of the Collatz graph, there is an amount of obvious structure in the road construction.
There is a more '''[http://www.teherba.org/fasces/oeis/collatz/roads.html elaborated example]''' for elements &lt;= 143248.
: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of roads can be seen.
====Road lengths====
====Road lengths====
* The lengths r<sub>1</sub> of the roads seem to be finite. The highlighted pairs of numbers &#x2261; 4 mod 6 are decreasing to the end of the road.
* The lengths r<sub>1</sub> of the roads seem to be finite. The highlighted pairs of numbers &#x2261; 4 mod 6 are decreasing to the end of the road.

Revision as of 21:28, 27 August 2018

Introduction

Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since many years it is unknown whether the final cyle 4 - 2 - 1 is always for any start value. 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.

Whenever we speak of numbers in this article, we mean natural, positive integer numbers (without 0).

References

Collatz graph

When all Collatz sequences are read backwards, they form the Collatz graph starting with 1, 2, 4, 8 ... . At each node n > 4 in the graph, the path from the root (4) can be continued

  • always to n * 2, and
  • sometimes also to (n - 1) / 3.

When n ≡ 0 mod 3, the path will continue with duplications only, since these maintain the divisibility by 3.

The 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 cylces.

Straightforward visualizations of the Collatz graph show now obvious structure. The sequences for the first dozen of start values seem to be rather harmless, but the sequence for 27 suddenly has 112 elements.

This article proposes

  • a rather regular structure for short portions of the Collatz graph, and
  • a corresponding algorithm which
    • combines these portions and
    • uses them to walk the graph in a systematic, predictable way
  • such that the reader could finally be convinced that the algorithm enumerates all numbers.

Steps

In order to walk back and forth in the Collatz graph, we need to describe how to move from one node in the graph to another. We will write

a s b

when use step s to move from node (element) a in the Collatz graph/sequence to node b. The following table lists the steps which we will use:

Name Mnemonic Direction Operation Condition Remark
h halve -> root b = a / 2 a ≡ 0 mod 2 next in sequence
t triple -> root b = 3 * a + 1 true next in sequence
m multiply root -> b = a * 2 true next up in tree
d divide root -> b = (a - 1) / 3 a ≡ 1 mod 3 next up in tree

Collatz roads

Motivation: Patterns in sequences with same length

When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, for example (from OEIS A070165):

142/104: [142 h  71 t 214 h 107 t 322 h 161 t 484 h  242 h 121 t 364 | 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
143/104: [143 t 430 h 215 t 646 h 323 t 970 h 485 t 1456 h 728 h 364 | 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
           +1  *6+4    +1  *6+4    +1  *6+4    +1   *6+4  *6+2      =    =  ...

The third line shows the relation between the values in the first two lines.

Informally, a road is a parallel arrangement of 2 pathes stemming from 2 sequences which have a common tail. In the example above, the road starts with an element ≡ 4 mod 6 (the 364 before the bar), followed by t/h and h/h steps, and the road then extends to the left as long as a characteristical, alternating sequence of pairs of steps ht - th - ht - th ... can be continued. In the example, the road can be continued to the left with 4 additional steps, but the construction then stops since there is no number q such that q * 3 + 1 = 62:

  q t | 62 h  31 t  94 h  47 t 142 h ...
126 h | 63 t 190 h  95 t 286 h 143 t ...
        +1  *6+4    +1  *6+4    +1  

Roads table R

For easier handling (e.g. in Excel) we list the roads for all starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are numbered r0, r1, r2 ...

r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 ...
start len dr0 mr0 mr2 mr3 mr4 dr5 dr6 mr7 mr8 dr9 dr10 mr11 mr12 dr13 ...
Δ6 Δ2 Δ12 Δ4 Δ24 Δ8 Δ8 3Δ8 3Δ48 3Δ16 3Δ16 9Δ16 9Δ96 9Δ32 9Δ32 ...
45182164511023
1033206401213
16353210642021
22474414882829958
283956181123637
3431168221364445
4091380261605253171063435117022237461415
4631592301846061
52317104342086869
58519116382327677251545051
64321128422568485
70323140462809293
764251525030410010133202

The roads contain 2 parallel, short, interrelated paths in the Collatz graph. Though the graph usually has a "chaotic" appearance, the roads exhibit quite some amount of regular structure.

There is a more elaborated example for elements <= 143248.

When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of roads can be seen.

Road lengths

  • The lengths r1 of the roads seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the road.
  • The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the road construction rule.
4 mod 54 10 mod 54 16 mod 54 22 mod 54 28 mod 54 34 mod 54 40 mod 54 46 mod 54 52 mod 54
5 3 3 4 3 3 n 3 3
  • At the starting values 4, 40, 364, 3280, 29524 (OEIS A191681) the road lengths ni have high values 5, 9, 13, 17, 21 which did not occur before. Those starting values are (9n+1 - 1) / 2, or 4 * Sum(9i, i=0..n).
  • The pattern of increasing and decreasing lengths is replicated when subsets of the rows (mod 9, 27, 81 ...) are regarded.

Coverage

The elements of the roads are strongly interconnected, and the table "obviously" shows all positive integers which are not multiples of 24:

r0 ≡ 4 mod 6 ≡ 4,10,16,22 mod 24
r3 ≡ 8 mod 12 ≡ 8,20 mod 24
r4 ≡ 2 mod 4 ≡ 2,6,10,14,18,22 mod 24
r5 ≡ 16 mod 24 ≡ 16 mod 24
r6 ≡ 4 mod 8 ≡ 4,12,20 mod 24
r2 ≡ 1 mod 2 all odd numbers

All odd multiples of 3 are contained in column r2. All multiples of 24 can be reached by duplicating them 3 times (3 m-steps).

So if we can show that we reach all start values ≡ 4 mod 6, we have a proof that all positive integers are reached.

Starting with 4, it seems possible that a continuous expansion of all numbers ≡ 4 mod 6 into roads would finally yield all roads up to some start value. Experiments show that there are limits for the numbers involved. Roads above the clamp value are not necessary in order to obtain all roads below and including the start value:

start value clamp value
4 4
40 76
364 2308
3280 143248

Subset table S

We may build derived table from the table of roads. We take columns r0 and r5 ff., and therein we keep the highlighted entries (those which are ≡ 4 mod 6) only, add 2 to them and divide them by 6. The resulting subset table S starts as follows:

s0  s1   s2   s3   s4   s5   s6   s7   s8   ...
 n  len  
 1   3    3    1    2
 2   1    7
 3   1   11
 4   3   15    5   10
 5   1   19
 6   1   23
 7   7   27    9   18    6   12    4    8
 8   1   31
 9   1   35
10   3   39   13   26
11   1   43
12   1   47
13   3   51   17   34
14   1   55
15   1   59
16   5   63   21   42   14   28
...

This table can be described by simple rules which are hopefully provable from the construction rule for the roads:

  • s2 is always s0 * 4 - 1.
  • When s2 ≡ 0 mod 3, the following columns s3, s4 ... are obtained by an alternating sequence of steps
    • si+1 = si / 3 and
    • si+2 = si+1 * 2,
    • until all factors 3 in s2 are replaced by factors 2.

Does S contain all positive integers?

The answer is yes. As above, we can regard the increments in successive columns:

ss ≡ 3 mod 4 half of the odd numbers
s3 ≡ 1 mod 4 other half of odd numbers
s4 ≡ 2 mod 8 ≡ 2,10 mod 16
s5 ≡ 6 mod 8 ≡ 6,14 mod 16
s6 ≡ 12 mod 16 ≡ 12 mod 16
s7 ≡ 4 mod 16 ≡ 4 mod 16
s8 ≡ 8 mod 32 8, 40, 72, ...
s9 ≡ 24 mod 32 24, 56, 88, ...
s10 ≡ 48 mod 64 48, 112, 176, 240 ...
s11 ≡ 16 mod 64 16, 80, ...

This shows that the columns s4 ... s7 contain all numbers ≡ 2,4,6,10,12,14 mod 16, but those ≡ 0,8 mod 16 are missing so far. The ones ≡ 8 mod 16 show up in s8 resp. s9, half of the multiples of 16 are in s10 resp. s11 but ≡ 0,32 mod 64 are missing, etc.

Since s2 contains arbitray high powers of 3, S has rows of arbitrary length, and for the missing multiples of powers of 2 the exponents can be driven above all limits.

Thus S contains all positive integers.

Can S be generated starting at 1?

We ask for an iterative process which starts with the row of S for index 1:

 1:    3    1    2

Then, all additional rows for the elements obtained so far are generated:

 2:    7
 3:   11

These rows are also expanded:

 7:   27    9   18    6   12    4    8
11:   43

Since we want to cover all indexes, we would first generate the rows for lower indexes. This process fills all rows up to s0 = 13 rather quickly, but the first 27 completely filled rows involve start numbers s0 up to 1539, and for the first 4831 rows, start values up to 4076811 are involved.