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Some time ago I came up with an idea of an infinitely-dimensional space that I personally call $\mathbb{P}^\infty$ or a prime powers space. It is probably known already, but I haven't found any references to it. I'd like to know is there any use for this besides just being a brain teaser.

Definition and properties:

  • Every positive rational number can be uniquely represented by a point in $\mathbb{P}^\infty$. Coordinates of such point correspond to prime factorization of the number. For example, 42 has the coordinates $(1,1,0,1)$ since $42 = 2^1 \cdot 3^1 \cdot 5^0 \cdot 7^1$, and $2/9$ is $(1,-2)$.
  • By definition, all prime numbers form a unit sphere of radius 1.
  • The space origin corresponds to number 1 since $p^0 = 1$ for every prime number $p$.
  • Amusingly, 0 does not map to any point in $\mathbb{P}^\infty$
  • Negative numbers, somewhat awkwardly, can be introduced using $i^2 = -1$

Questions:

  • Was this object known before?
  • Is there a way to analytically extend it beyond rationals?
  • What lies between the grid lines, say, between $2^1$ and $2^2$?
  • What is the meaning of translation or rotation of a figure in such space?
  • What would be the metric?
  • It is known that some geometric facts are formulated more easily in a proper coordinate system. For example, many functions are trivially defined using polar coordinates. Are there any mathematical facts that could be conveniently formulated using this space?
Korvin
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1 Answers1

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This is essentially a special case of the construction of fractional ideals of a Dedekind domain, in this case $\mathbb{Z}$. Fractional ideals can be defined for rings of integers of number fields $K$ which generalize $\mathbb{Z}$, such as the Gaussian integers $\mathbb{Z}[i]$. More geometric (in the sense of algebraic geometry) examples are also possible; see divisor.

Abstractly the group of fractional ideals of a Dedekind domain $D$ is always free abelian on the prime ideals of $D$ (this is a version of unique factorization for ideals, which is one of the defining features of Dedekind domains), so each fractional ideal can be specified by a list of coordinates giving the exponent of each prime ideal. This use of "prime coordinates" is a neat way to see very concretely the effects of operations such as gcd and lcm, but I'm not aware of any extremely common name for it.

Qiaochu Yuan
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