I need help in seeing if I can turn an idea into a proof, or if I am merely ruining Math with my attempt at solving this. I recently watched a youtube video on the Collatz Conjecture. The Conjecture is simple enough:

Start with a positive number n and repeatedly apply these simple rules:

- If
n= 1, stop.- If
nis even, dividenby 2.- If
nis odd, multiplynby 3 and add 1.

Now the way I approached the matter was first by switchingninto binary format. So lets say we start out with 59 I turn it into 111011. So following the rules you would multiply by 3 or 11.

00111011

x 11

_____________

00111011

+01110110

_____________

10110001

or 177

Then you add 1 and get 10110010 or 178.

Next since it is even you would divide by 2 and get 89 or 1011001.

Long story short, repeating this you end up with 1.

So here comes the part where I need help getting this to the proof stage. But the basic concept is this, no matter what numbernyou start out with it can always be represented in binary, and no matter how large the first number of the string will be a 1, and if it is odd it ends with a 1, even it is a 0. The division by 2 in binary of an even number is simply truncation of the string. So you end up with the first 1 and the last 1. This causes the string to be multiplied by 3. This multiplication causes the shifting of the string to the left and keeps a 1 at its right end, thus the addition of 1 comes in and frees up the string to be truncated by the division by 2. Ultimately the end result of the odd rule and 3nwill yield a string that is nothing but 1s and thus the addition of 1 will give a string with 1 and the rest 0, which is then truncated down to just 1.

To further show that this holds true, you can change the Odd rule to 3n+/-mwheremis any odd number less than 3n.The end result of this change of rules will still yield 1, as the first part of the string is 1 and the last will be 0 due to adding an odd number.

Sorry that this is a bit rambling. Like I said, I kinda need help getting this into a way that is much easier to understand. It all makes sense to me, but maybe I'm not seeing a flaw in this understanding.