Hi all.

I've had this idea - a conjecture in the field of Number Theory - for a few years now. The conjecture is rather simple, as were the logical steps that I made in order to infer it, so I would have assumed that it had already been suggested in the past. Nevertheless, I have not been able to find any piece of evidence that it had (possibly due to the difficulty of "phrasing" it into Google's search engine).

I would appreciate your opinion of the following:

- Are you familiar with this theorem in any way?
- Is it eligible to be stated as an open conjecture in Number Theory?

My conjecture can be stated in any of the following ways:

- No set S ⊂ { (3n+2)/(2n+1) │ n∈N } exists such that ∏Si is a power of 2
- No set S ⊂ { (3n+2)/(2n+1) │ n∈N } exists such that ∏Si is integer
- No multi-set S ⊂ { (3n+2)/(2n+1) │ n∈N } exists such that ∏Si is integer

Reminder:

- In a set, no element can appear more than once
- In a multi-set, any element may appear more than once

In simple words: Take any group of numbers from the series {5/3, 8/5, 11/7, 14/9, 17/11, 20/13, ...}. Calculate the product (multiplication) of these numbers - the result will never be an integer number.

Note:

In its weakest form (#1), my conjecture is sufficient for proving that there are no cyclic sequences in the '3n+1' conjecture (the proof for that is pretty simple, but I am not including it here because it is not the main purpose of my question). I believe that my conjecture also holds in its strongest form (#3).

Thank you very much for your time.