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 n is even, divide n by 2.
- If n is odd, multiply n by 3 and add 1.
Now the way I approached the matter was first by switching n into 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.
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 number n you 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 3n will 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 +/- m where m is 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.