While reading this thread Is 1 a prime number?, I recalled that The Fundamental Theorem of Arithmetic (FTA) which says that every positive integer greater than 1 can get written uniquely as a product of only primes except for order of the factors. What does "uniqueness" here mean?
I ask, because the way I interpret uniqueness, if uniqueness exists, it means that there exists only one way (except for notation) of writing such a product of primes. E. G. for 6, we can only write (2#3) or (3#2), with "#" indicating multiplication. Those expressions only differ in the order of factors so the theorem applies there.
However, this doesn't always work. For example, say we consider 30 as the integer to get written as a product of primes. Then we can write 30 as ((2#3)#5) as well as writing (2#(3#5)) (note I don't see how 2#3#5 is clear as a unique product, but only that the two meaningful choices which would make it as a shorthand for a product equal each other). The order of factors ((2#3)#5) does not change when passing to (2#(3#5)), nor do we change the notation, so we have two different ways of "writing" 30, which perhaps becomes even more transparent to see if we require that all products get written in prefix or suffix notation (e.g. 30 can get written as ##235, or #2#35, so long as we understand the numerals appropriately). So, does uniqueness fail for FTA, or what exactly does uniqueness mean here? If uniqueness fails, what does the Fundamental Theorem of Arithmetic try to say? Does it mean something along the lines that for any positive integer c greater than 1 there exists a unique bag or multiset b such that no matter which way you take the product of two members of the bag to yield a result r, take the product of r and a member of the bag not used so far, and repeat this process until you have no members of the bag left, you will obtain the integer c?
Addendum: What happens to the Fundamental Theorem of Arithmetic if you require that all products get written via either a prefix notation scheme ("Polish" notation), or a suffix notation scheme ("Reverse Polish" notation)?