On first post and second post parts, I set-up the mathematical context to prove Collatz conjecture.

## Cluster analysis

To prove Collatz sequence conjecture, we need some extraneous analysis. Reusing results of digit mutation proof, we know that the transition path from one number to another is driven by its unit digit, and that the transitions are always the same for one digit.

### Cluster definitions

I name cluster a group of transition path from one number to another.

#### Definition of cluster 1

cluster 1 is made of numbers with unit digit being part of set S_{1} = { , , , }.

#### Definition of cluster 2

cluster 2 is made of numbers with unit digit being part of set S_{2} = { , }.

#### Definition of cluster 3

cluster 3 is made of numbers with unit digit being part of set S_{3} = { , , , }.

**NOTA BENE**: The union of the clusters, cover all the digit unit cases. So, cluster analysis results will be true whatever the input number considered.

## Analysis of cluster 2

### Collatz sequence analysis

#### Collatz sequence analysis on numbers ending with digit

These numbers can be expressed using canonical form as 10 t + 9. These numbers are .

So, I apply the rule of Collatz sequence. This brings the calculus formulae 30 t + 28.

This formulae can be rewritten under canonical form as 10 3t + 8. Therefore, the result will have a digit .

##### Focus on tens of

Let start from a number ending with digit 9, using ω suite.

ω_{n + 1} = 3 ω_{n} + 1 by definition

ω_{n + 1} = (3 ω_{n} + 3) + (1 - 3) = 3 ( ω_{n} + 1) - 2

ω_{n + 1} = 3 (100 h + 10 t + 9 + 1) - 2 = 100 3h + 10 3t + 30 - 2 = 100 3h + 10 3t + 20 + 8 = 100 3h + 10 (3t + 2) + 8

Therefore, input number has t tens, and output number has 3t + 2 tens.

#### Collatz sequence analysis on numbers ending with digit

These numbers can be expressed using canonical form as 10 t + 8. These numbers are .

So, I apply the rule of Collatz sequence. This brings the calculus formulae 5 t + 4.

This formulae can be rewritten under conditions

- t is so, t = 2p and so the canonical form is 10 p + 4, leading to digit
- t is so, t = 2p + 1 and so the canonical form is 10 p + 9, leading to digit

### Computation analysis

#### Computation on numbers ending with digit with tens

Let’s consider the cases where tens are 0, 2, 4, 6, or 8.

υ_{n} |
||
---|---|---|

n | n + 1 | n + 2 |

09 | 28 | 14 |

29 | 88 | 44 |

49 | 148 | 74 |

69 | 208 | 104 |

89 | 268 | 134 |

All the n + 2 results, are out of the cluster. This proves by calculus, that any tens of a number ending with digit drives the suite to another cluster in just 2 steps.

#### Computation on numbers ending with digit with tens

Let’s consider the cases where tens are 1, 3, 5, 7, or 9.

υ_{n} |
|||
---|---|---|---|

n | n + 1 | n + 2 | comment |

19 | 58 | 29 | even tens for number ending with digit |

39 | 118 | 59 | loop to case 59 of this table |

59 | 178 | 89 | even tens for number ending with digit |

79 | 238 | 119 | loop to case 19 of this table |

99 | 298 | 149 | even tens for number ending with digit |

So, the numbers 19, 59, and 99 will exit the cluster in 4 steps.

So, the numbers 39, and 79 will exit the cluster in 6 steps.

#### Conclusion of computation analysis

I have shown that whatever the number ending by digit , we can predict the number of steps before exiting the cluster and reaching systematically a number ending by digit . Here is a summary table

υ_{n} |
||||
---|---|---|---|---|

t | u | computation sequence | expansion factor | |

t is | ωη | 3 / 2 = 1.5 | ||

t is ∈ { , , } | ωη ωη | 9 / 4 = 2.25 | ||

t is ∈ { , } | ωηωηωη | 27 / 8 = 3.375 |

The computation sequence is just showing the order of application of Collatz sequence.

So the conclusion is whatever the input number ending with digit , the Collatz sequence will drive its sequence to a number ending with digit in 2, 4 or 6 steps.

Moreover, starting with number ending with digit , owning an number of tens will bring to the sequence from a number ending with digit . In this case, the expansion factors shown above should be divided by 2.

The cluster 2 is therefore acting as a multiplier of its input. It neither diverges nor converges. It just creates an output number greater than its input number.