Since every natural number can be represented as $a=2^{n_1}3^{n_2}5^{n_3}7^{n_4}\cdots p_k^{n_k}\cdots$ it makes sense to represent natural numbers by vectors, using the properties of logarithms:

$$\log a=n_1 \log 2+n_2 \log3+n_3 \log5+\cdots$$

This space appears to be similar to the usual Euclidean space if we extend it to an infinite number of dimensions.

If we allow negative coordinates, we can also put all rational numbers in this space. For example, here is part of the plane $(\log 2, \log 3)$:

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Does this space have any application in number theory? If it is studied, then how is it usually defined? Are the usual Cartesian vector dot product and the usual Euclidean norm used? Or does it make sense to use a different norm (for example, taxicab norm)?

  • $\begingroup$ You'll have to be clearer about what "this space" means. In the representation, then $n_i$ are integers, and only finitely many of them are non-zero. To be a vector space, then $n_i$ have to be allowed to be in a field. $\endgroup$ – Thomas Andrews Jul 9 '16 at 23:13
  • $\begingroup$ @ThomasAndrews, I agree. To define the properties of the vector space I can allow $n_i$ to be real. $\endgroup$ – Yuriy S Jul 9 '16 at 23:17
  • $\begingroup$ You mean the positive rationals. This observation is essentially equivalent to unique prime factorization, and in that sense is implicitly used all the time in number theory. I'm not aware of people making use of any norms though. $\endgroup$ – Qiaochu Yuan Jul 10 '16 at 5:41

There is a generalization of vector spaces called "modules" which allow any ring to serve as scalars. When you use the integers as the ring of scalars, a "module" is the same thing as a "abelian group".

The group of 'factorizations' is indeed a free abelian group, which is the kind of abelian group that behaves most similarly to a vector space.

Factorizations are indeed important in number theory. More generally, rather than the rationals you might consider number fields or even global fields. You would then consider things like prime ideals or places instead of prime numbers.

Formally taking logarithms like you are is somewhat superfluous — what you're doing is mainly just changing the notation of the group operation to $+$ so that it's easier to think about it in terms of linear algebra.

It can indeed be useful to extend to real coefficients rather than merely integer coefficients. e.g. after restricting to a finite set of primes, number theorists like to view the group of factorizations as a lattice contained in the corresponding vector space $\mathbb{R}^n$ and use geometric methods to study things.

The most natural norm to take here is a weighted $L^1$ norm

$$ \left\| \sum_{n=0}^{\infty} a_n \log p_n \right\| = \sum_{n=0}^{\infty} |a_n| \ln p_n $$

This way, the norm of the factorization of an integer is precisely the natural logarithm of the magnitude of that integer. More generally, if $a$ and $b$ are relatively prime nonzero integers, then the norm of the factorization of $a/b$ is simply $\ln|a/b|$.

  • $\begingroup$ But wouldn't it defeat the point of representing numbers in a different way, if we give them the same norm as usual? The idea was to get something not just from factorization, but from this 'vector space representation' $\endgroup$ – Yuriy S Jul 10 '16 at 0:09
  • $\begingroup$ @You're: One main point of representing numbers in a different way is to allow the use of new techniques to study the things we're interested in. That works best when we can actually express the things we're interested in to the things the new techniques talk about! $\endgroup$ – user14972 Jul 10 '16 at 5:29

I asked myself the same a while ago, and I have an interesting answer: that representation is, for instance, the key argument of Rusza's proof for the existence of an infinite (almost optimal in terms of density) Sidon set.


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