My Mathematical Theories

Possible Approach For Collatz Conjecture

Collatz conjecture states that

choose a number greater than 1
Let that number be the x

-If x is an even number then divide it by 2
-If x is an odd number then multiply it by 3 add the product to 1

The answer will be the new number(x) and do the same method.

It states that all number will always end up to 1.

It can be formulated.

             x mod 2=0  x/2
f(x) ={
             x mod 2=1  3x+1

But is fⁿ(x)=1?
n is the number of steps you did until you reach 1, if it will alaways reach 1

Example
Let x=5
f¹(5)=3(5)+1
        =16
f²(5)=(16)/2
        =8
f³(5)=(8)/2
        =4
f⁴(5)=(4)/2
        =1

As you can see above in 4 we have reached 1

The other way to formulate it is
           x mod 2=0 x/2
f(x) ={
            x mod 2=1  (3x+1)/2

Since 3x+1 will always end up even number if x is odd number.

Using the formula itself we can formulate that

Let x sub 0 an even number
such that
x sub 0/2ⁿ=x   (x is odd number)

y[k] =y sub k

fⁿ(x) =(3ⁿ+3ⁿ¯¹x+3ⁿ¯²2^{y[1]}+3ⁿ¯³2^{y[1]+y[2]}+...+3¹2^{y[1]+y[2]+...+y[n-2]}+2^{y[1]+y[2]+...+y[n-1]})/2^{y[1]+y[2]+...+y[n]}

The first approach we might conclude is
Proving 3ⁿ+3ⁿ¯¹x+3ⁿ¯²2^{y[1]}+3ⁿ¯³2^{y[1]+y[2]}+...+3¹2^{y[1]+y[2]+...+y[n-2]}+2^{y[1]+y[2]+...+y[n-1]}
is equal to
2^{y[1]+y[2]+...+y[n]}

And also including finding the relation of x and n

Like
x|n

2|1 if x is 2 then n is 1.we only did is one step, the 2/2
We will use the original formula the.(3x+1 without /2)

3|7
4|2
5|5
6|8
7|16
8|3
9|19
10|6
11|14

The other approach is finding the numbers in the set of col
Let col() is the set of numbers in which using the formula it will end up to 1

Like finding if x=col() such that x is possitive integer greater than 1

It is obvious that
2ⁿ=col()
Using this approach can we solve now the unsolved math problem for many years?