The Collatz Conjecture is a famously unproven problem in mathematics, but I was thinking of a slight modification, and whether or not a proof of this different form is trivial.
Here is a statement of the original problem;
Take a starting integer n and apply the following operations; multiply by 3n + 1 if odd, divide by two if even. Take the resulting number and feed back into the algorithm. After a finite number of steps the algorithm will reach 1.
Here is a modified statement;
For any integer n you can always apply some combination of multiplying by 3n + 1 and dividing by 2 to get to 1.
The difference is that you don't explicitly define what to do in the case when a number is even or odd. Of course, you can only divide by 2 when the number you are working with is even, but in this statement, taking an even number and multiplying by 3n+1 follows the rules so, in some ways, the Collatz conjecture is a special case of this form.
Anyways, has anyone heard of a proof of something like this and if not, would a proof tell us anything interesting about the Collatz conjecture itself?
Thanks!