TITLE: A conjecture in Number Theory
QUESTION [9 upvotes]: 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.

REPLY [39 votes]: Even conjecture 1. is false: For $S=\{(3n+2)/(2n+1)\;|\;n=9, 12, 14, 27, 41\}$ your product is $8$.