Collatz Streetmap: Difference between revisions

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imported>Gfis
proof of coverage reduced to = 4 mod 6
imported>Gfis
clamp
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Each road contains 2 short pathes 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.
Each road contains 2 short pathes 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.
====Road length====
===Road length===
* The lengths r<sub>1</sub> of the roads seem to be finite.
* The lengths r<sub>1</sub> of the roads seem to be finite.
* The lengths show a repeating pattern: n<sub>1</sub>,3,3,4,3,3, n<sub>2</sub>,3,3,4,3,3 n<sub>3</sub>,3,3,4,3,3 ...
* The lengths show a repeating pattern: n<sub>1</sub>,3,3,4,3,3, n<sub>2</sub>,3,3,4,3,3 n<sub>3</sub>,3,3,4,3,3 ...
* At the starting values 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]) the road lengths n<sub>i</sub> have high values 5, 9, 13, 17, 21 which did not occur before. Those starting values are (9<sup>n+1</sup> - 1) / 2, or 4 * Sum(9<sup>i</sup>, i=0..n).
* At the starting values 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]) the road lengths n<sub>i</sub> have high values 5, 9, 13, 17, 21 which did not occur before. Those starting values are (9<sup>n+1</sup> - 1) / 2, or 4 * Sum(9<sup>i</sup>, i=0..n).
* The pattern of increasing and decreasing lengths is replicated when subsets of the rows (mod 9, 27, 81 ...) are regarded.
* The pattern of increasing and decreasing lengths is replicated when subsets of the rows (mod 9, 27, 81 ...) are regarded.
====Coverage====
===Coverage===
The elements of the roads are strongly interconnected, and the array "obviously" shows all positive integers which are not multiples of 24:
The elements of the roads are strongly interconnected, and the array "obviously" shows all positive integers which are not multiples of 24:
{| class="wikitable"
{| class="wikitable"
| r<sub>0</sub> &#x2261; 4 mod 6
| r<sub>0</sub> &#x2261; 4 mod 6
| &#x2261; 4,10,14,18,22 mod 24
| &#x2261; 4,10,16,22 mod 24
|-
|-
| r<sub>3</sub> &#x2261; 8 mod 12
| r<sub>3</sub> &#x2261; 8 mod 12
Line 129: Line 129:


So if we can show that we reach all start values &#x2261; 4 mod 6, we have a proof that all positive integers are reached.
So if we can show that we reach all start values &#x2261; 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 &#x2261; 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:
{| class="wikitable"
! start value
! clamp value
|-
| 4 
| 4
|-
| 40
| 76
|-
| 364
| 2308
|-
| 3280
| 143248
|}

Revision as of 20:41, 24 August 2018

Introduction

When all Collatz sequences are read backwards, they form a graph starting with 1, 2 ..., hopefully without cycles (except for 1,2,4,1,2,4 ...). At each node n in the graph, the path starting at the root (4) and with the last node n can in principle be continued to 2 new nodes by a

  • "m"-step: n * 2 (which is always possible), or a
  • "d"-step: (n - 1) / 3 (which is possible only if n ≡ 1 mod 3).

When n mod 3 = 0, the path will continue with m-steps only, since the duplication maintains the divisibility by 3.

References

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 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]
           +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]

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

d m m d m d m d ... 
m m d m d m d m ...

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:

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).

For better handling (e.g. in Excel), the roads for all starting values 4, 10, 16, 22 ... 4+6*n are listed as rows in an array. The cells in 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

There is a more elaborated example for elements <= 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.

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 pathes 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.

Road length

  • The lengths r1 of the roads seem to be finite.
  • The lengths show a repeating pattern: n1,3,3,4,3,3, n2,3,3,4,3,3 n3,3,3,4,3,3 ...
  • 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 array "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 are obtained by 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