Straight line intersects geometric sequence

[adar]

Summary:: Two parallel lines (same slope) - one intersects the y-axis, and the other doesn't.
Trying to find the intersection of either with a given geometric sequence.

The lines are:
y=mx
y=mx+1
The values on one or the other of the lines - but not both simultaneously - are to be completely divisible by 2 (end result to be 1). To find out which, I consider an intersection of these with the geometric sequence below.

The geometric sequence:
x_n=2ⁿ

Is this a workable approach?
Have I made this too murky?
Please let me know.

Thanks in advance.

[Moderator's note: moved from a technical forum.]

 

[jedishrfu]

What do you mean by one line intersects the y-axis and the other doesn’t and yet they are parallel lines?

 

[PeroK]

Have I made this too murky?
Please let me know.

Thanks in advance.

I don't understand what you are trying to do.

 

[adar]

What do you mean by one line intersects the y-axis and the other doesn’t and yet they are parallel lines?

Same slope, different lengths.

 

[adar]

I don't understand what you are trying to do.

All positive integers [1→∞) are involved.
Those which are completely divisible by 2 in their original form could conceivably be represented on the mx line (as an idea).
Those which are not at all divisible by 2 must be transformed by adding 1, thus becoming mx+1 (if the above idea holds).
All the numbers which are divisible completely by 2 (both the mx and the mx+1) enter a special domain.
The others must undergo the mx+1 transformation until they become completely divisible by 2.
When the division by 2 of a given number is stopped by some component in the number (e.g., in 6=2⋅3 the component is 3), that component must undergo the transformation.
The process continues until the newly obtained number is completely divisible by 2.

It is evident that the division is assuredly uninterrupted - the only way to enter the new domain - if the divisor is anywhere in the sequence x_n=2ⁿ

The gist is:
Will all the integers - those which are not already completely divisible by 2 - end up being integers completely divisible by 2 after any number of mx+1 transformations?
If mx and mx+1 are acceptable geometric representations of the classes of integers, I was looking for a way to find their intersections with the sequence mentioned above.

Brute force shows that quintillions of integers do enter the domain after the transformations; sometimes huge number of transformations!
But do all of [1→∞) end up there?!
And how to prove it?

 

[PeroK]

It seems you have an iteration:

1) Start with a positive integer $n$.
2) Take out all factors of $2$ to get $n_2$, which must be odd.
(If $n_2 =1$, then stop.)
3) Add $1$ to $n_2$ and repeat from step 1.

The question is whether you always reach $1$ at some stage. This is equivalent to getting a power of $2$ at some stage.

It's similar in some ways to the Collatz conjecture:

https://en.wikipedia.org/wiki/Collatz_conjecture

Except, in this case, starting with an odd integer, adding $1$ and dividing by $2$ (or higher powers) will always give a lower number. So, we are bound to get to $1$ eventually.