TITLE: Is there a "canonical" representation of integers using numbers other than primes?
QUESTION [7 upvotes]: Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, $S$ of integers.
When $S$ is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which this is true. 
EDIT: As the answers show, this obviously cannot be done. What if we relax the integer condition, i.e. can there be any other canonical representation of positive integers using complex numbers?

REPLY [2 votes]: When you relax the integrality condition you start to pick up other solutions - for instance, the set of square roots of primes, $S=\{\sqrt{p}:p\in P\}$, is such that every number $n$ has a unique factorization $n=s_1^{e_1}s_2^{e_2}\ldots$ in terms of the elements of $S$; just consider the breakdown $n=p_1^{d_1}p_2^{d_2}\ldots$ and then set $e_1=2d_1$, etc.  On the other hand, this gives up existence : it's no longer the case that every sequence $e_1, e_2, \ldots$ of whole numbers corresponds to a unique integer; instead we have the additional requirement that all the $e_i$ be even.
More generally, any breakdown of primes into products of (non-integral) real factors will give you a solution: if $p_1 = t_{11}\cdot t_{12} \cdot \ldots\cdot t_{1m}$, $p_2 = t_{21}\cdot t_{22}\cdot\ldots\cdot t_{2n}$, and so on, then obviously any factorization of a number $n$ into products of $p_i$ extends to a (unique) factorization of $n$ into products of the $t_{ij}$.  What's more, any unique factorization of all numbers into real 'factors' $x_i$ has to come from this kind of breakdown, because obviously the prime numbers all need unique factorizations, and the factorization of a number $n$ into primes then 'extends through' the factorization of those primes into a factorization of $n$ in terms of the $x_i$, which by definition must be the factorization of $n$.  So you don't get anything particularly interesting by removing the non-integrality condition, just 'refinements' of the factorizations you already had.