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I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?

I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $\prod p_i^{\alpha_i}$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $\alpha_1$ units from the origin along the $p_1$ axis, then $\alpha_2$ units parallel to the $p_2$ axis, then $\alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $\alpha_i=0$ for all $i$.

Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.

Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $\prod p_i^{\alpha_i}$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^{\alpha_1}=p_2^{\alpha_2}$.

I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?


marked as duplicate by Bill Dubuque number-theory Dec 21 '18 at 2:52

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    $\begingroup$ What a fascinating idea! $\endgroup$ – DonielF Dec 20 '18 at 3:57
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    $\begingroup$ You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers. $\endgroup$ – Anguepa Dec 20 '18 at 3:59
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    $\begingroup$ @YiFan: Which group do you mean by "[t]his..group"? $\endgroup$ – user 170039 Dec 20 '18 at 4:07
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    $\begingroup$ You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles. $\endgroup$ – Lubin Dec 20 '18 at 4:08
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    $\begingroup$ It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas $\endgroup$ – J.G. Dec 20 '18 at 11:02

For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^c\ldots$ is represented by the tuple $(a,b,c,\ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.

Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $\sqrt 2=2^{1/2}$ can only be expressed that way.

If you allow real exponents you no longer have a unique point for each number. You could represent $\sqrt 5$ as the point $\frac 12$ on the $5$ axis, but you could also represent it as the point $\frac 12 \log_2(5)$ on the $2$ axis or many other ways.

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    $\begingroup$ Would it be a field? I'm not sure it's closed under addition - e.g. can you express $\sqrt2+\sqrt3$ as a product of rational powers of primes? $\endgroup$ – Carmeister Dec 20 '18 at 8:49
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    $\begingroup$ @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $\sqrt[k]{q}$ for some positive integer $k$ and positive rational number $q$. $\endgroup$ – Servaes Dec 20 '18 at 10:34
  • $\begingroup$ @Carmeister: you are correct. I will remove that. $\endgroup$ – Ross Millikan Dec 20 '18 at 14:46
  • $\begingroup$ if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold? $\endgroup$ – WorldSEnder Dec 20 '18 at 19:23
  • $\begingroup$ @WorldSEnder: I believe so. $\endgroup$ – Ross Millikan Dec 20 '18 at 19:55

I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:

1   2   4
5   10  20
25  50  100
125 250 500

The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.

This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.

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    $\begingroup$ This connection is discussed in this answer and its comments. $\endgroup$ – Mark S. Dec 20 '18 at 18:58

In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?

What about cryptograpy?

Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...

Good luck


The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $\mathbb{F}_2$.

You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 \pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $\mathbb{F}_2$ with dimension equal to the size of the factor base.

Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 \equiv y^2 \pmod n$ by combining the relations. The reason we work over $\mathbb{F}_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.

With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $\gcd(n, x + y)$ or $\gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).

The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $\mathbb{Z}$.


Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:

    2   3   5   7
first plot (trivial)
1   0   0   0   0

second plot
2   1   0   0   0
2   0   x   0   0
2   0   0   y   0
2   0   z1  z2  0

third plot
2.5 .   .   .   .
2.5 .   .   .   .
2.5 .   .   .   .

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