Proving the Collatz Conjecture with Binaries Numbers

The objective of this article is to demonstrate the Collatz Conjecture through the Sets and Binary Numbers Theory, in this manner: 2 + 2+...1. This study shows that there are subsequences of odd numbers within the Collatz sequences, and that by proving the proposition is true for these subsequences, it is subsequently proven that the entire proposition is correct. It is also proven that a sequence which begins with a natural number is generated by a set of operations: Multiplication by 3, addition of 1 and division by 2. This set of operations shall be called “Movement” in this study, and may be increasing when n=1, and decreasing for n ≥ 2. The numbers in 2 form generate decreasing sequences in which the 3n+1 operation does not occur. One of the important discoveries is how to generate numbers in which the 3n+1 operation only occurs once and how to generate numbers with a minimum quantity of increasing movements that are the numbers of greater “orbits” (Longer sequences that take longer to reach the number one). The conclusion is that, as the decreasing numbers dominate as compared to the increasing ones, the statement that the sequence is always going to reach the number 1 is true.


Introduction
In this article, the Theory of Sets and Binary Numbers will be used, with the ED [Portuguese acronym] (Written by Definition) method, to investigate the Collatz conjecture through the results obtained. It is shown that the orbit of each number is determined by its binary form. This paper also demonstrates how to obtain numbers in which the 3n+1 operation does not take place at all, takes place only once, or in which this operation appears at least "n" times.
The Collatz Conjecture, or 3n+1 problem, was formulated in 1937 by German mathematician, Lothar Collatz. It is a mathematical assumption which is thought to be true, but has yet to be proven or rejected.
The Collatz Conjecture asserts that, by performing the following operations: begin with a natural number. If this number is even, divide by 2. If it is odd, multiply by 3 and add 1. After this, a new number is obtained and the process repeated. Lothar Collatz conjectured that, by pursuing these operations, one will always arrive at the number 1.
Observe the example of number 10, in which we arrive at the sequence: 10, 5, 16, 8, 4, 2, 1. This sequence is called the Hailstone Sequence, and has its operations closed whenever it reaches the number 1. The elements that make up this sequence are called the orbit. As such, the orbit of number 10 has 7 elements.
Portuguese researcher, Tomás Oliveira e Silva, explored a number of assumptions, starting at number 1 and surpassing the number 19x2 58 . He did not come across any case in which the result did not reach 1.
Based on this Conjecture, Filho [1] proposes the Collatz Function C (n) f:N→N, defined as follows: C n if n is even 3n 1 if n is odd (1) According to Filho [1], the sequence of numbers generated through this process until reaching the number 1 is called Orbit Number n. This is a sequence obtained by recurrence. According to Lima et al. [8], "A sequence is recursively defined when obtained by a rule that allows for the calculation of any term in function of the immediate precursor (s). For Lima et al. [ ], "For a sequence to be perfectly determined, knowledge of the first terms is also necessary." Cz (n) refers to the orbit of a natural number, n > 1, the sequence obtained by recursively applying the Collatz function, starting from the natural number n, by successively applying the function C (n) until the sequence reaches the number 1.
Cz (n) is a way of identifying the orbits without ambiguities among other mathematical subjects. Using the first and last letter of the name Collatz allows for easy identification of the orbits in any language. Observe three orbits: Cz (6)  This problem contains a challenge which is to understand the orbit of each number. How can it be explained that numbers of such similar values produce such different sequences? The most likely possibility is that the orbit be explained in conjunction with the demonstration of the Conjecture. Another possibility is that, after the demonstration of the Conjecture, the orbit becomes the next challenge. However, perhaps to the surprise of mathematicians researching this problem, at the end of this chapter we shall see that what determines the orbit of each number is its own binary form.
This research began with the reading of an article in "Revista Cálculo", Osone [3], in which the conjecture was disclosed. After reading the article, I searched the Internet for earlier attempts to prove the conjecture. This was when the work of American mathematician Lagarias [4] stood out, in which he organized all advances related this conjecture up to that point (2013). The next step was to seek tools that could be used to solve the problem in the literature on Numbers Theory, from such authors as Carvalho [5], Chaves [6], Coutinho [7], Neto [8], Hefez [9] and Scheinerman [10].
The work of both Lesieutre [11] and Carnielli [12], which generalizes the conjecture for other numbers, particularly divisors other than 2, was also consulted. However, generalization proposals were not used in this study though other of the authors' ideas have been considered.
In his book about the last Fermat theorem, Singh [13] shows how conjectures are important for the development of Mathematics, as studies that do not reach the objective produce important contributions to science. What actually took place during this research was the discovery of a new Diophantine equation and a Cryptographic system that is useful for entertaining students and motivating the study of Numbers Theory. These discoveries are described in Santos' book [14].
For Stewart [15], the conjectures can be formulated in a way that is both simple and easy to understand for those that have mastered Basic Mathematics. Others are so complex that only specialists are able to understand their formulation. However, the advantage of the simple-formula conjectures is that they are "democratic"; they invite students of any age to study Mathematics. Andrew Wiles began trying to solve the last Fermat Theorem while he was still a child, and fulfilled his dream after receiving his doctorate. That's why it is interesting to present the simple formulation of the conjectures, such as the Collatz, to the students.
Sing [13] and Stewart [14] were important in the choice of strategies and also in this author's motivation, showing that, by describing the attempts of other researchers, with their successes, failures and advances, it was worthwhile to try solving the Collatz Conjecture. And, after the choice of strategies and research of previous results were done, an attempt was made at a demonstration, the results of which are presented below.

Proposition
If a natural number, n, is presented as 2 n , the CZ (n) set does not have the 3n+1 operation.

Proposition
If n is a natural number with n=3 or n≥5, and n≠8, and Cz (16) is contained in Cz (n).
Demonstration: By definition, the last term of the sequence produced by Collatz operations is 1. As 3n+1>1, the previous operation was a division by 2, thus the penultimate number is 2. Similarly, as 3n+1>2, the number preceding 2 is 4.
For 3n+1= 4 => n=1. This is ridiculous because 1 is the last number in the sequence, so the term preceding 4 is 8.
Based upon the arguments above, the proposition is true. The Cassini [16] graph demonstrates this proposition.

Proposition
The last elements of the Collatz sequence prior to the number 1 form a Collatz subsequence.
This property allows the Collatz sets to be written more elegantly, and also tests whether the conjecture is true for an untested number more quickly. This property appears in the work of other mathematicians, such as Lagarias (2013) Assuming that a 3 =b 4 , therefore, the result is:

Proposition
If n is an odd natural number, Cz (n) is a subset of natural Cz (2 p n) ∀ , i.e., Collatz sets of odd numbers are subsets of even numbers. Demonstration: In accordance with the fundamental theorem of arithmetic, a natural even number may be written as a product of primes including 2 p . As such, if K is a natural even number, it can be written as follows: K=2 p x3 q x5 t x....V z . It is understood that 3 q x5 t x....V z = M=> M is a number formed by the product of odd numbers, and so, it is odd. Applying the Collatz operations, the result is Cz This property bring about an interesting conclusion: that to prove or disprove the conjecture, one just needs work with odd numbers.

Proposition
If two odd numbers are used, A and B, where B is a multiple of A, it cannot be said that Cz (A) is a subset of Cz (B). Example: Cz (3)

Collatz Operations with Binaries
The advantage of doing Collatz operations in base 2 is that, in this format, it is possible to see numerous hidden properties in base 10. This method of writing the binary will be referred to as ED [Portuguese acronym -Writing by Definition], and each power of 2 n is called a "term". Before beginning the Collatz operations, the base number 2 is transformed into its ED equivalent.
With this form of writing numbers in any base, Santos [14], a variety of properties and information are obtained, which are not visible in traditional Hindu-Arabica script.
Therefore, ICz (n) is a subsequence of odd Cz (n) numbers. Examples; The advantages of ICz (n) with respect to Cz (n) may be observed through the graphing of each sequence. As an example, we consider the Cz (7) with ICz (7) graphs: Cz (7) Graph  Note how the ICz (7) graph is much more regular, improving observation of each number's orbit.

Collatz Movements
As has already been defined, Collatz Movement is the 3n+1 operation which is then divided by 2 n in order to obtain the ICz (n) set. This movement will also be classified in accordance with its divider, 2 n . The exponent "n" will be the degree of movement. As such, if the division is by 2, the movement will be of the first degree. If divided by 2 3 , it will be a third-degree movement. Therefore, ICz (13) has a third-degree movement as well as a fourth-degree movement.
Second-degree movement is what results most frequently and, hence, is known as "Monotonous Motion".

Analysis of Collatz Movement
Collatz movement can be divided into three phases: multiplication, addition and division.
In the multiplication stage, each 2 n term produces a new 2 n+1 term that will be known as the "New Term", and which causes the quantity of the terms to double.

Proposition
The first-degree movement is increasing. All other movements are decreasing. Demonstration: We take a natural number, "n", which is greater than two. As such, we have: 3n+1>2n Dividing the two terms by 2, we have: (3n+1):2 > n. Note that (3n+1):2 is the first-degree movement. Supposing that "p" is a natural number which is greater than, or equal to, one, then we have: 3n+1< 2 p x2n And so, dividing the two terms by 2 p x2 we have: (3n+1):2 p x2 < n (3n+1):2 p+1 < n Once again, note that (3n+1):2 p+1 is movement of a degree which is greater than, or equal to, two.

Proposition
Monotonous movement occurs whenever the penultimate term has a degree which is greater than second. If the degree of the penultimate term is second, we have a strong reduction. If the degree is first, we shall have an increasing movement.

Proposition
If the last two terms of a number are 2+1, the next movement of said number is augmentative.

Theorem
If the exponents of a number's "N" terms are in descending order, this number will have at least "N" augmentative movements which will be the first n movements of ICz. Demonstration: We take the number 2 n +2 n-1 +2 n-2 ....2 3 +2 2 +2+1.
When the exponents of two or more of a number's terms with a difference of just one unit in relation to the previous or following term, these terms will be called "aligned terms" or "term alignment".

The Effects of Multiplication and Addition on the Terms
As a new term with an exponent that is greater than the original term is created in the multiplication step, the terms will become "closer", i.e., the difference between its terms decreases and, in the addition, they can join to form a single term. This result of multiplication and addition operations will be called "Fusion", and is independent of the kind of movement.
There are two types of fusion. The first is when the exponents of the terms have one unit of difference.
(2 n+2 +2 n-1 +2 2 ):(2 2 ) = 2 n +2 n-3 +1 The other fusion is when there are two units of difference between the exponents of the terms which precede a term with a single unit.
A very important consequence of these operations is that the first terms undergo more fusions than the last ones because, with every multiplication, a new term is created that has one more unit than the term suffering multiplication. Additionally, every two moves, this term undergoes a fusion which creates a term with an even larger. The process repeats until the new terms undergo fusion with the terms ahead of them.
Another important effect of the multiplication and addition operations is the change in parity. This explains the sudden changes in their orbits.
between the second last and third last terms is only one unit, 2 n-1 + 2 n-2 e 2 n-5 +2 n-6 . This is key to understanding the orbits because it causes new alignments between the latter terms, which leads to new augmentative movements. This, in turn, makes the orbit of some numbers so complicated because the sequence decreases, and then increases again as the latter terms suffer repeated realignments. However, as has been shown in the previous item, the multiplication and addition operations produce terms with differing parity. As such, after an alternation of augmentative movements followed by monotonous movements, there will be a sequence of monotonous movements initiated by the penultimate term with an even exponent, which will end in a strong reduction. If, after the strong reduction, the penultimate term has and odd exponent, there will be monotonous movements followed by augmentative movement. Should it be even, there will be monotonous movements followed by a strong reduction.

Discussion
Mathematicians who have studied the Collatz Conjecture over the past 80 years have come to the conclusion that the mathematical knowledge available to them, including computers, was not up to the task of demonstrating. It took a new kind of number, the binary ED, to unravel this puzzle.
The ED numbers also produced other discoveries which can be found in Santos' book. It has only been published in Portuguese, to date, and does not contain the results laid out in this article. This, in fact, is the great advantage of researching conjectures in any area of mathematics. In addition to being fun and challenging, it also produces new mathematical knowledge that can be used in other areas.
Demonstrations of the generalizations created by other researchers are still lacking, and should remain for other mathematicians. They should be all that much easier with the knowledge found in the results of this research.

Conclusion
The factor that ensures that the Collatz Conjecture is a Theorem, is that the decreasing movements predominate in relation to the increasing movements. Even in a number formed by terms of odd exponents and a large initial alignment, the effect of multiplication and addition operations will cause the terms to have exponents with increasing differences. Additionally, the fusions will change the parities of the exponents. This leads to monotonous movements followed strong reductions.
Even if someone produces a very complicated number, using all of the knowledge about sockets and binary numbers, this sequence will have decreasing orbits with monotonous movements and strong reductions that, together, will predominate in relation to the increasing movement. This will always lead to the sequence ending in the number 1.