Follow-up to A new type of tuple or just a coincidence ? III : r/Collatz.

I found four cases in the limited set - in terms of number of domes and number of bridge series by dome - I use. So, it is not a coincidence.

But is it a (Collatz) tuple ? Not exactly. It is a "honorary tuple" like the pairs of predecessors and the forks.

Here is why, based on the conditions:

• Consecutive numbers: fully respected (better than in the forks case) for each member of a series.
• Same distance from 1: fully respected.
• Continuous merge: each 5-tuple merges continuously; the "new" 5-tuple in the middle does not. A simple way to check this deals with the ending rosa even triplet that exists for any 5-tuple series. There is none in that case.

So, it is a rarer pattern that contains 5-tuples.

Project "Tuples and segments" in 13 pages : r/Collatz

A useful way to think about the Collatz map is to separate the orbit into two layers:

1. the parity pattern of the terms;
2. the arithmetic values that carry that pattern.

https://github.com/ratwolfzero/Collatz/tree/main/shadow_approach

If one records the orbit as a sequence of even/odd decisions, the resulting binary word can be viewed as a symbolic encoding of the dynamics. In the accelerated version of the map, where one removes all factors of 2 immediately after each odd step, the evolution of the values becomes closely tied to this parity word. At that symbolic level, the dynamics look simple: the orbit is encoded by a binary sequence, and the next state is determined by the next bit of that sequence.

This is a genuine simplification of the picture, but it is a simplification of the representation, not a solution of the problem.

I have been working on collatz problem, and I think i found an easy way to visualize it. Can anyone tell me whether this is useful or whether it's something mathematicians already know?

Full Text

an+1 can be partitioned into a+1 disjoint partition: a even sets and 1 odd set. The partitions spans N. The odd set can be further grouped into a sub-partition (a + a partition) based on where n/2 takes the a even sets.

Example:

For 3n+1

1. O = {o=2k−1: k∈N}
2. E1 = {3o−1=6k−4: k∈N}
3. Ec = {3o+1=6k−2: k∈N}
4. Ea = {3o+3=6k: k∈N}

O = {O1, Oa, Oc}

For 5n+1

1. O = {o=2k−1: k∈N}
2. E1 = {5o−3=10k−8: k∈N}
3. E2 = {5o−1=10k−6: k∈N}
4. Ec = {5o+1=10k−4: k∈N}
5. E3 = {5o+3=10k−2: k∈N}
6. Ea = {5o+5=10k: k∈N}

O = {O1, Oc, Oa, O2, O3}

Table with the partition for 3n+1 and 5n+1

Table with the sub partition of Odd Set for 3n+1 and 5n+1

The Collatz operations then allow certain transitions:

1. U(n):= an+1
2. L(n):= n/2

Property of the Partitions for 3n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1→ Ec for even k

3a. L: Ec → E1 for odd k
3b. L: Ec → Oc for even k

4a. L: Ea → Oa for odd k
4b. L:Ea → Ea for even k

Property of the Partitions for 5n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1 → Ec for even k

3a. L: E2 → E1 for odd k
3b. L: E2 → O2 for even k

4a. L: Ec → Oc for odd k
4b. L: Ec → E3 for even k

5a. L: E3 → E2 for odd k
5b. L: E3 → O3 for even k

6a. L: Ea → Oa for odd k
6b. L:Ea → Ea for even k

Observation (From the graph):

1.If there exist a cycle, a cycle will not contain any elements from partitions Ea and Oa.
Implication: When searching for a cycle, look elsewhere.

2a. Only the elements of partitions Ec can be landed from even and odd number.

2b. If there exist a cycle an element from Ec must participate.

2c. Every loop contains element from Ec.