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:


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



A conjecture in Number Theory
Even conjecture 1. is false: For $S=\{(3n+2)/(2n+1)\;|\;n=9, 12, 14, 27, 41\}$ your product is $8$.


