OEIS/3x+1 Problem
Introduction
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 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.
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
- Gottfried Helms: The Collatz-Problem. A view into some 3x+1-trees and a new fractal graphic representation. Univ. Kassel.
- Klaus Brennecke: Collatzfolgen und Schachbrett, on Wikibooks
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.
When m ≡ 0 mod 3, the path will continue with duplications only, since these maintain the divisibility by 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 be concerned with this cycle, and we start with node 4, the root.
Graph Operations
Following Trümper, we use abbreviations for the elementary operations which transform a node (element, number) in the Collatz graph:
| 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 | step | +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. In the opposite direction, the s operation replaces a factor 2 in m by 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.
Segment Construction
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 C[i,j], the so-called segment directory. One row i of C is constructed according to the following table:
| Column | Operation | Formula | Condition | Sequence |
|---|---|---|---|---|
| C[i,1] | 6 * i - 2 | 4, 10, 16, 22, 28, ... | ||
| C[i,2] | C[i,1] µµ | 24 * (i - 1) + 16 | 16, 40, 64, 88, 112, ... | |
| C[i,3] | C[i,1] δµµ | 24 * (i - 1) / 3 + 4 | i ≡ 1 mod 3 | 4, 28, 52, 76, 100, ... |
| C[i,4] | C[i,2] σ | 48 * (i - 1) / 3 + 10 | i ≡ 1 mod 3 | 10, 58, 106, 134, ... |
| C[i,5] | C[i,3] σ | 48 * (i - 7) / 9 + 34 | i ≡ 7 mod 9 | 34, 82, 130, 178, ... |
| C[i,6] | C[i,2] σσ | 96 * (i - 7) / 9 + 70 | i ≡ 7 mod 9 | 70, 166, 262, 358, ... |
| C[i,7] | C[i,3] σσ | 96 * (i - 7) / 27 + 22 | i ≡ 7 mod 27 | 22, 118, 214, 310, ... |
| C[i,8] | C[i,2] σσσ | 192 * (i - 7) / 27 + 46 | i ≡ 7 mod 27 | 46, 238, 430, 622, ... |
| C[i,9] | C[i,3] σσσ | 192 * (i - 61) / 81 + 142 | i ≡ 61 mod 81 | 142, 334, ... |
| ... | ... | ... | ... | ... |
The first column(s) C[i,1] will be denoted as the left part of the segment (or of the whole directory), while the columns C[i,j], j > 1 will be the right part.
- All nodes in the segment directory are of 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.
- All segments have a finite length.
- At some point the σ operations will have replaced all factors 3 by 2.
- All nodes in the right part of a segment have the same kernel.
- This follows from the formula for columns C[i,1..3], and since the σ operation maintains this property.
- All nodes in the right part of a segment are different.
- 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 (right part of segment starting with 40, kernel 1):
160 = 6 * 33 * 20 - 2 52 = 6 * 32 * 20 - 2 106 = 6 * 32 * 21 - 2 34 = 6 * 31 * 21 - 2 70 = 6 * 31 * 22 - 2 22 = 6 * 30 * 22 - 2 46 = 6 * 30 * 23 - 2
- There is no cycle in a segment.
Segment Lengths
The segment directories are obviously very structured. The lengths of the compressed segments follow the pattern
4 2 2 4 2 2 L1 2 2 4 2 2 4 2 2 L2 2 2 4 2 2 ...
with two fixed lengths 2 and 4 and some variable lengths L1, L2 ... > 4. At the starting values 4, 40, 364, 3280, 29524 (OEIS A191681), the segment lengths have high values 4, 8, 12, 16, 20 which did not occur before. Those starting values are (9n+1 - 1) / 2, or 4 * Sum(9i, 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:
| 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.
- 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:
- All numbers of the form 6 * n - 2 occur exactly once in the right part C[i,j], j > 1 of the segment directory.
Connectivity
We now aim to build a (directed) tree with nodes of the form 6 * n - 2. We take the segment starting at 4:
4 -> 16 -> 4 -> 10
and for any node x in the right part (x > 4) we connect the segment starting with x with the current segment. We repeat this process for all additionally connected segments.
4 -> 16 -> 4 -> 10
| |
V V
64 -> 256 40 -> 160 -> 52 -> 106 -> 34 -> 70 -> 22 -> 46
...
- The process does not reach a cycle.
- A cyle 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.
- The process will never stop.
- All nodes added by a segment are different from those already in the tree.
- The process reaches all segments.
- (This is not shown so far.)
Collatz Tree
Once the coverage and the connectivity and cycle-freeness of the segments is established, it is easy to switch back to the full Collatz graph and prove that it is a tree (except for the trivial cycle 4-2-1). We feaze the segment, and we list the intermediate results of the operations in two separate paths:
6*i-2 µ µµ µµδ µµδµ µµδµδ µµδµδµ ...
δ δµ δµµ δµµδ δµµδµ δµµδµδ ...
We list the resulting subpaths in two rows, and we call this subset graph a detailed segment (in contrast to the compressed segments described above), and a detailed segment directory can easily be created by a Perl program.
- This straightforward construction of the detailed segments introduces some unpleasant duplications of nodes in the right part. Later we simply suppress such duplicated nodes.
From a segment start of the form 6 * i - 2 we reach the numbers x according to the following table:
| Number | Operation | Remark |
|---|---|---|
| x ≡ 1, 3, 5 mod 6 | (6 * i - 2) δ = 2 * i - 1 | all odd numbers |
| x ≡ 0 mod 6 | (3p) µq | odd multiples of 3 times powers of 2 |
| x ≡ 8 mod 12 | (6 * i - 2) µ | 8, 20, 32, 44, ... |
| x ≡ 2 mod 12 | (6 * i - 2) δµ if i ≡ 1 mod 3 | 2, 14, 26, 38, ... |
Together this shows that we can reach all numbers ≡ 0, 1, 2, 3, 5 mod 6 which are not contained in the right part of the compressed segment directory.
The connections between the segments via the nodes 6 * i - 2 remain unaffected.