A new way of looking at the Collatz conjecture

[Lourens]

A new way of looking at the Collatz conjecture
Lourens Nicolaas Jacobus Engelbrecht

Abstract. The Collatz conjecture is named after a mathematician Lothar Collatz who wrote the conjecture in 1937. The conjecture which is an
unsolved problem in mathematics, also known as the 3n + 1 conjecture, is a conjecture in mathematics that explains a sequence defined
as follows:
Starting with any positive integer, the starting integer can be an even number or an odd number.
Consider a number n, if the number n is even, divide it by 2.
If the number n is odd, triple it and add one.
Further, it states that regardless of the choice of n, after some iterations of the conjecture, the number no matter what value of a positive n is chosen, the sequence from the number chosen projecting to peak and lower values will always reach 1.
Following the theme, here in this article, the sequence of the projections are discussed and also shown is how numbers are isolated in certain fixed
projections, until the final number 1 is reached.

Mathematics Subject Classification (2010). 00A05; 00A06; 00A69

Keywords. Congruence, Collatz conjecture.

1. Introduction
Mathematical theorems are important building blocks for applications in mathematics and computing. Other uses are for general research, cryptography, encryption, and many more applications in everyday use. Although no consensus exists yet for the proof of the Collatz conjecture, the search for a more efficient proof is of the utmost importance to mathematics which may contribute to solving other known problems and applications. The sequence of numbers involved in this conjecture is sometimes referred to as hailstone numbers because of the different properties of each number in multiple descents and ascents before reaching the number 1.

The famous mathematician Paul Erdos said that mathematics may not be
ready for such problems as the Collatz conjecture, and Jeffrey Lagarias, another famous mathematician stated in 2010 that it is an extraordinarily difficult problem, completely out of reach of present-day mathematics. This article is specifically based on a novel theorem. So by adding three novel theorems, we devised a new method to analyze the iterations of numbers in the Collatz conjecture through a 3 Column approach which explains the properties of the patterns including peak values. An illustration also shows the projection and interconnection of odd and even numbers within and between the 3 columns.
In this proof, the main notions which we are using are defined as follows:
To explain this conjecture, we construct a three-column structure of numbers which will help us in following the projections. The three-column structure
is as follows.
C1 : [1 4 7 10 13 16 19 22 25 · · · ] T, Odd and Even
C2 : [2 5 8 11 14 17 20 23 26 · · · ] T, Odd and Even
C3 : [3 6 9 12 15 18 21 24 27 · · · ] T, Odd and Even.
where T is the transpose of the array, odd represents an odd number in the specific column, even represents an even number in the specific column.
• Column 1 numbers once divided by 3 always has .333333 as decimal
• Column 2 numbers once divided by 3 always has .66666 as decimals
• Column 3 numbers once divided by 3 always have .0000 as decimals.
We now fix some notations.

We set C1 mean a number in Column 1,
C1-0 means an odd number in Column 1,
C1-E means an even number in Column 1.

Let C2 mean a number in Column 2,
C2-0 means an odd number in Column 2,
C2-E means an even number in Column 2.

Let C3 mean a number in Column 3,
C3-0 means an odd number in Column 3,
C3-E means an even number in
Column 3.

2. Main section
Lets us start with the three column structure of numbers:
C1 : [1 4 7 10 13 16 19 22 25 · · · ] T, Odd and Even
C2 : [2 5 8 11 14 17 20 23 26 · · · ] T, Odd and Even
C3 : [3 6 9 12 15 18 21 24 27 · · · ] T, Odd and Even.
It is easy to observe that the:
C1 class once divided by 3 always has .333333 as decimal,
C2 class once divided by 3 always has .66666 as decimals and
C3class is exactly divisible by 3.
We know that in Collatz conjecture, if n is odd, then we have two operations: magnification by 3 and then translation by one. In other words, n is
projected to an even number. And if n is even, then we divide it by two until it projects to a column as an odd number.
We now discuss the following six projections:

Case 1:
The Collatz number of the C3-E subclass is always projected to become a C3-O subclass number with no return path to C3-E. The process may end with
one iteration or multiple iterations. For example:
186 ∈ C3-E reaches to 93 ∈ C3-O in only iteration.
But 24 ∈ C3-E in the first iteration reaches 12 ∈ C3-E, 12 to 6 ∈ C3-E, and finally 6 to 3 ∈ C3-O.

Case 2:
The Collatz number of the C3-O subclass operates with magnification by 3 followed by translation of one and are always projected after only one iteration
to become a C1-E number with no return to C3. For example:
following the example in case 1, 3 ∈ C3-O is projected to 10 ∈ C1-E.
Also, one can easily see the following one iteration projections
129 ∈ C3-O projected to 388 ∈ C1-E
219 ∈ C3-O projected to 658 ∈ C1-E
2373 ∈ C3-O projected to 7120 ∈ C1-E
We also observe that a number of C3 classes cannot ever be a projection of a number of C1 classes or C2 classes.
Moreover, C3 numbers are thus now isolated outside of C3 after the above iterations to receive any projections back from C1 and C2.

Case 3:
The Collatz projection of C1-E subclass numbers are the numbers from C2- E subclass or C2-O subclass, and requires only one iteration. For example:
(i): 76 ∈ C1-E is projected to 38 ∈ C2-E,
ii): 544 ∈ C1-E is projected to 272 ∈ C2-E,
(iii): 34 ∈ C1-E is projected to 16 ∈ C2-O,
(iv): 538 ∈ C1-E is projected to 269 ∈ C2-O.

Case 4:
(i): 31 ∈ C1-O is projected to 94 ∈ C1-E,
(ii): 223 ∈ C1-O projected to 670 ∈ C1-E. C3 number.

Case 5:
By the Collatz conjecture, every number of C2-E subclass is projected to a number of C1-E subclasses or to a number of C1-O subclasses after one iteration
only. For example;
(i) : 140 ∈ C2-E projected to 70 ∈ C1-E,
(ii) : 788 ∈ C2-E projected to 394 ∈ C1-E,
(iii): 1478 ∈ C2-E projected to 739 ∈ C1-O,
(iv) :146 ∈ C2-E projected to 73 ∈ C1-O.

Case 6:
Using the Collatz conjecture every number of C2-O subclass is projected to anumber of C1-E subclasses after one iteration only. For examples;
(i) : 41 ∈ C2- O projected to 124 ∈ C1-E,
(ii): 143 ∈ C2-O projected to 430 ∈ C1-E,
(iii):1067 ∈ C2-O projected to 3202 ∈ C-E.
Moreover, no C2-E or C2-O number ever projects to become a C3 number. From the above discussion, we can easily state the following:

Theorem 2.1.
The Collatz conjecture projects every number of C3-E subclass to a number of C3-O subclass with no return path to C3-E. ​Also every number
of C3-O subclass is projected to a C1-E subclass after one iteration only with no return to C3.
​
From the above theorem, we can state the following corollary:
Corollary 2.2. The C3 numbers are isolated outside of C3 after the above iterations to receive any projections from C1 or C2.

Theorem 2.3.
The Collatz conjecture projects numbers of C1-E subclass to numbers of C2-E subclass or to numbers of C2-O subclass after one iteration
only. Also, every number of C1-O subclass projects to stay in C1 to become a C1-E number after one iteration only. And a C1-E or C1-O number can
never project to become a C3 number.

Theorem 2.4.
The Collatz conjecture projects every number of C2-E subclass to a number of C1-E subclasses or C1-O subclass after one iteration only. Also,
every number of C2-O subclass is projected to a number of C1-E subclasses after one iteration only. And a C2-E or C2-O number can never project to
become a C3 number.

From the above cases, we can easily state that because of the set patterns in the 3 column structure we observe that:
The number values of C3 are after a few iterations totally out of the equation by being projected to C1-E with no return.
Any number through the hailstorm chaos reaches the final {4, 2, 1} loop.
The Collatz conjecture is proven as true, as long as the numbers stay to the three column structure with its unique properties.

3. Conclusion
It is easy through this novel approach and its observations that proof of the Collatz conjecture is not intractable as thought by studying the pattern and
relationships between numbers in the three-column structure namely:

All C3-O and C3-E numbers are finally projected to C1-E.
All C2-O numbers are finally projected to C1-E.
All C2-E numbers are finally projected to C1-E and C1-O.
C1-O and C2-E are the outlets for C1-E ( The storm’s heart) from the isolated
receivers C3-O; C2-O and in a lesser way C2-E and C1-O.
This approach indicates that all orbits are bounded.

4. Conflict of interest statement
The author declares that there is no conflict of interest in this paper. The author declares that there are no other financial relationships with any organizations that might have an interest in the submitted work.

References
[1] J.C. Lagarias, The Ultimate Challenge: The 3x + 1 Problem, American Mathematical Society, 2010.
[2] Hailstone Number, MathWorld. Wolfram Research.
[3] Guy, Richard K., E17: Permutation Sequences. Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 336–7. ISBN 0-387-20860-7. Zbl
1058.11001, 2004.
[4] Lagarias, A heuristic argument, 1985.

Lourens Nicolaas Jacobus Engelbrecht
e-mail: engelbrechtlourie@gmail.com