TITLE: Mapping natural numbers into prime-exponents space
QUESTION [27 upvotes]: Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$, 
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example, 
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$ 
because $123456789=2^0 3^2 \cdots 3607^1 \cdots 3803^1$.
So every number in $\mathbb{N}$ is mapped to a point in an arbitrarily high dimensional space.
This mapping has the property that addition of the vectors corresponds to multiplication of
the numbers.
My (extremely vague!) question is: Does this viewpoint helps gain any insights into the 
structure/properties of natural numbers?
Do line/planes/curves in this space mark out numerically interesting regions?
Perhaps allowing real or complex numbers?
The numbers in $\mathbb{N}$ fill this infinite-dimensional space very sparsely.
This is (very!) far from my research expertise, so any comments/references/links would be appreciated.

REPLY [2 votes]: A friend of mine and me were discussing this possibility some years ago. This vectorial space with infinite dimension is what we called the "encoded space". There are some interesting facts, for instance, prime numbers in $\mathbb{N}$ are in the sphere of radius 1 in the enconded space. Also, you can encode far beyond natural numbers. In fact, you can encode the whole set of $\mathbb{Q}$, rational numbers if you write negative numbers inside the coordinates of the vector. But the thing is that you can even encode any root of a rational number if you add the rational numbers inside the vector. For instance:
$$ 12 = 2^2·3 \rightarrow  (2, 1, 0, 0,...) = (2, 1 \rangle $$
$$ \frac{12}{7} = 2^2·3·7^{-1} \rightarrow (2, 1, 0, -1 \rangle $$
$$ \sqrt[7]{\frac{12}{5}} = (2^2·3·5^{-1})^{\frac{1}{7}} =2^{\frac{2}{7}}·3^{\frac{1}{7}}·5^{-\frac{1}{7}} \rightarrow (\frac{2}{7}, \frac{1}{7}, -\frac{1}{7} \rangle $$
Also, some transcendental numbers can be encoded if you add to the vector some irrational algebraic numbers:
$$ 2^{\sqrt{2}} \rightarrow (\sqrt{2}\rangle $$
There is no problem if you try to encode these kind of numbers, BUT the really hard problem arrives when you try to encode the usual summation between real numbers. For instance, you can encode $\frac{1}{2}$ and also $\frac{\sqrt{5}}{2}$, but you cannot encode this:
$$ \frac{1}{2} + \frac{ \sqrt{5}}{2} $$
I've been trying to find a way to encode this but dealing with summation codification is difficult.

I think that the most interesting thing is that it's easy to prove that the exponent-vectors and the encoded space behave as a vector space. It can easily proved (we did some years ago). The most important implication of this is that you can take all the theorems from linear algebra and apply them to the natural numbers or the rational numbers.
SPOILER: we try to get the equivalent theorems from algebra to $\mathbb{N}$ and $\mathbb{Q}$ and it was a little bit disappointing (pretty obvious facts).