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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.

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@KennyTM: Yes, of course, any extensions are welcome.! I am especially interested in what natural curves in this space imply for real numbers... –  Joseph O'Rourke Aug 20 '10 at 20:15
This map looks like a Godel encoding. I think you will want to define this as a map into an infinite dimensional space rather than "arbitrarily high dimensional" (by simply putting zeroes after all nonzero powers of primes in the vector). I have heard of using some topology (Zariski/positive-information topology?) on this space but never geometry... interesting question! –  Tom Stephens Aug 20 '10 at 20:16
Just a naïve comment in the midst of the knowledgeable responses, but I thought it worth mentioning that your map is a monoid isomorphism from the positive integers with multiplication to the countably infinite direct sum of the nonnegative integers with addition. This is just another way of saying that there are infinitely many primes and that each natural number has a unique prime factorization. –  Jonas Meyer Oct 31 '10 at 18:50
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2 Answers

up vote 9 down vote accepted

In algebraic number theory such rings of valuation vectors or adèles or repartitions play a key role in globally structuring local information for the global study of number fields. In particular, the topological properties of such adèle rings imply key results in algebraic number theory such as the structure of unit groups, finiteness of the class number, various results at the foundation of the Riemann-Roch theorem, etc. For example, see Tate's below review of a paper by Iwasawa (esp. the penultimate paragraph).

MR0053970 (14,849a) 10.0X
Iwasawa, Kenkichi. On the rings of valuation vectors. Ann. of Math. (2) 57, (1953). 331--356.

Let $K$ be a finite algebraic number field, or an algebraic function field of one variable over a finite constant field. The ring $R$ of valuation vectors over $K$ has the following properties: (1) $R$ is a semi-simple commutative ring with unit element 1; (2) $R$ is locally compact, but neither compact nor discrete; (3) $R$ has a subfield $K$ containing 1, such that $K$ is discrete in $R$ and the residue class space $R/K$ is compact.

In the first half of his paper the author proves conversely that any topological ring having these properties is the ring of valuation vectors over a field $K$ of the type described above. The main tool used is the notion of the norm $N(\sigma,G)$ of an automorphism $\sigma$ of a locally compact group $G$, which is the factor by which the Haar measure in $G$ is stretched by $\sigma$. It is multiplicative in $\sigma$; $N(\sigma,G)=N(\sigma,H)N(\sigma,G/H)$ if $\sigma H\subset H$; and $N(\sigma,G)=1$ if $G$ is compact or discrete. If $F$ is a nondiscrete locally compact field, then the function $N(\alpha,F)$ for $\alpha\in F^\ast$ gives the "normed" valuation of $F$. The intersection of all closed maximal ideals $M$ of $R$ is 0, and $N(x,R)=\prod_MN(x,R/M)$ for regular $x\in R$. In particular, if $\xi\in K^\ast$, then from (3), $N(\xi,R)=1$, so that the valuations $N(\xi,R/M)$ of $K$ satisfy the ``product formula.'' From this it follows by more or less familiar methods that $K$ is an arithmetic field and $R$ its valuation vector ring. The next section contains a proof that valuation vector rings do enjoy properties (1), (2), and (3).

Finally, the author shows that the two central theorems of algebraic number theory can be proved directly from the topological properties of $R$. The first of these is the self-duality of $R$ with respect to $K$, on which the analog of the Riemann-Roch theorem for number fields rests. If $\chi$ is any non-trivial additive character of $R$ vanishing on $K$, then the map $x\rightarrow\chi(xy)$ is an isomorphism of $R$ onto its character group, and $\chi(x\xi)=1$ for all $\xi\in K$ if and only if $x\in K$. The essential step in the proof is to show that the kernel of the map $x\rightarrow\chi(xy)$ is a compact ideal of $R$, and is therefore 0. The second theorem states the compactness of the group of idèle classes of norm 1, which is equivalent to the finiteness of class number and unit theorem. The idèle group $J$ is the multiplicative group of regular elements of $R$. It is a locally compact group in the topology for which the convergence of $a_i$ in $J$ means the convergence of $a_i$ and $a_i{}^{-1}$ in $R$. The map $a\rightarrow N(a,R)$ is a homomorphism of $J$ with kernel $J_1$, and the compactness of $J_1/K^\ast$ is proved by a Minkowski-type argument using Haar measure.

Throughout the paper, the case of function fields over non-finite constant fields is treated in parallel, using linear compactness instead of compactness.
Reviewed by J. T. Tate

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What a terrifically detailed response to my ridiculously imprecise question! Adèle rings Indeed! –  Joseph O'Rourke Aug 20 '10 at 21:34
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This is a really broad question, so I'll just point you to two closely related notions. (About which, I should admit, I know very little.)

In the function field setting, the objects you're mapping into go by the name of divisors. Divisors are an important tool in algebraic geometry and the theory of Riemann surfaces, e.g. this is how people talk about Riemann-Roch. Back in the number field setting one can still define divisors and even ask for an arithmetic analogue of Riemann-Roch, which is described in the relevant chapter of Neukirch (I don't have it on hand at the moment). This type of stuff is the domain of Arakelov theory.

Alternately, instead of just taking the exponents of the prime factorization you could embed the integers into their adele ring; this is, for example, the modern way to do class field theory.

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A wonderfully rich response, well beyond what my vague question deserves. I will learn much from exploring your links. Thanks so much! –  Joseph O'Rourke Aug 20 '10 at 20:42
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