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Frederick Williams

Joined: 19 Nov 2005
Posts: 97

Posted: Fri Jul 07, 2006 1:20 am    Post subject: Re: My prime research.

"doctormungmung@gmail.com" wrote:
 Quote: This is my first post here, so excuse any etiquette i accidentally miss. My method was to treat each prime as a dimension in an infinite dimensional space. So ... the number 2 would correspond to the vector [1,0,0,0 .....] the number 3 would correspond to the vector [0,1,0,0 .....] the number 5 would correspond to the vector [0,0,1,0 .....] and as all integers are primes or composites, you would be able to represent every integer in this space.

Represented how? I mean, consider 90 = 2 3^2 5, how does that get
represented in terms of [1,0,0,0,...], [0,1,0,0 ...] and [0,0,1,0 ...]?

Also, if by integers you mean ...,-2,-1,0,1,2,..., you need to represent
negation as well.

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Remove "antispam" and ".invalid" for e-mail address.
William Elliot
science forum Guru

Joined: 24 Mar 2005
Posts: 1906

Posted: Fri Jul 07, 2006 1:39 am    Post subject: Re: My prime research.

On Thu, 6 Jul 2006, doctormungmung@gmail.com wrote:

 Quote: This is my first post here, so excuse any etiquette i accidentally miss.

Patrick Hamlyn
science forum beginner

Joined: 03 May 2005
Posts: 45

Posted: Fri Jul 07, 2006 3:05 am    Post subject: Re: My prime research.

"doctormungmung@gmail.com" <doctormungmung@gmail.com> wrote:

 Quote: This is my first post here, so excuse any etiquette i accidentally miss. My method was to treat each prime as a dimension in an infinite dimensional space. So ... the number 2 would correspond to the vector [1,0,0,0 .....] the number 3 would correspond to the vector [0,1,0,0 .....] the number 5 would correspond to the vector [0,0,1,0 .....] and as all integers are primes or composites, you would be able to represent every integer in this space.

Not immediately apparent what you mean here. Are you limiting yourself to unit
vectors (with plus or minus sign, else you couldn't represent '4'), or are you
allowing any positive integers or any reals? I'll assume you mean non-negative
integers, because if you allow reals you could just write every number X as
[X/2,0,0,0,0 ...]

 Quote: also i think the map could work with the positive rationals if 1 mapped to [0,0,0,0 ......]

Not sure what you mean here either. How would you represent 2/3?

Also, any system with a non-zero quantity mapped to the zero vector is probably
going to break in many other ways. 0+0 equals 1+1 for example. You just broke
all arithmetic in the system.

 Quote: anyway, the reason this seemed like a good idea, is that each point that has a distance 1 away from zero is prime...

This would support my assumption that only integers can be used in each
dimension, otherwise this statement would not be true. Numbers of the form
P1*sqrt(1/2) + P2*sqrt(1/2) all can be represented by a vector having distance 1
from the origin, where P1, P2 are distinct primes.

 Quote: ... All other numbers are composite. So I was hoping to find a pattern in the distance away from zero, as we go up the number line. I made graphs on www.tonydoran.com. And it seems that there is some order to the graph. The way I like to think the distance is kind of like a measure of how prime the number is, if the number has large distance, it is less prime, and a smaller one is more prime.... (it does sound wierd).

It would seem that most composites have more than one representation in this
system, as indeed do most primes. Even if you specify that the only
representation for any number is the one that is closest to zero, some numbers
have several representations equally close to zero. In fact it's not obvious to
me how to easily find the 'closest to zero' representation for very large
numbers.

 Quote: What I am also curious is to find ways to describe how this space gets populated. Also i am interested about finding other metrics that are easy to calculate in many dimensions.

'Dimensionality' could be one. Primes require one dimension to describe, all
composites can be represented by two dimensions (usually not the representation
closest to zero). If we limit the domain of allowable values for each dimension,
some composites might be representable only by using three or more dimensions.
For some such domain (eg +/- 1) it might be interesting to define the sequence
of numbers N1,N2,N3... being 'the smallest number which has dimensionality n'.

Alternately, using the domain 'positive integers' for each dimension the
'closest to zero' representation might require three or more dimensions.

The 'minimum dimension sequence' for such a system would start (did this in my
head so errors & omissions excepted) (2,8,27,...)

In other words '2' is the smallest number which can be described minimally with
a single dimension, 8 is the smallest number which has a minimum representation
which has two dimensions [0,1,1] (we use this as shorthand for
[0,1,1,0,0,0,...]) and I think 27 is the smallest which requires three
dimensions, eg [0,1,1,0,0,0,0,1] or [0,1,0,1,0,0,1] or [0,1,0,0,1,1] (ie 3+5+19
or 3+7+17 or 3+11+13). After that it gets tricky, since you will eventually have
to look at vectors involving digits larger than 1. In fact 4 and 6 are such a
numbers, I think their minimum representations are [2] and [0,2] respectively.

It might even be interesting to define 'each dimension has a domain such that
every number has a unique minimum representation in the system'. Not sure
whether this works though.

Clarify the rules a bit, so I know I'm on the right track...

I looked at your site, and you appear to have not considered the 'minimum
distance from zero' representation, eg 27 you represent as [0,7].
jasen
science forum beginner

Joined: 28 Jun 2006
Posts: 16

Posted: Fri Jul 07, 2006 9:25 am    Post subject: Re: My prime research.

On 2006-07-06, doctormungmung@gmail.com <doctormungmung@gmail.com> wrote:

 Quote: What I am also curious is to find ways to describe how this space gets populated. Also i am interested about finding other metrics that are easy to calculate in many dimensions.

angle would be one, may not be very intersting in this case though.

--

Bye.
Jasen
doctormungmung@gmail.com1
science forum beginner

Joined: 06 Jul 2006
Posts: 2

Posted: Fri Jul 07, 2006 1:56 pm    Post subject: Re: My prime research.

let me clarify the way i thought this should work

each integer can be written as :

p(1)^a * p(2)^b * p(3)^c ...... where p(n) is the n-th prime

i want to represent each number as vector whose values would be:
[a,b,c.........]

so in the case of the number 6:
2^1 * 3^1 * 5^0 *7^0 .....
[1,1,0,0,.......]

or in the case of the number 27:
2^0 * 3^3 * 5^0 * 7^0 ......
[0,3,0,......]

or in the case of the number 2450:
2^1 * 3^0 * 5^2 * 7^2 * 11^0 ......
[1,0,2,2,0,0,......]

this would imply the 1 = [0,0,0,0,.....]
and 2/3 = [1,-1,0,0,0,0......]
Irrationals are not going to make sense here because they are not going
to have multiple representations

Adding vectors in this system is equivalent to multiplying the numbers
that they represent.
A = [1,0,2,0......]
B = [0,2,0,0......]
then A + B = [1,2,2,0.....]
A represents 2^1 * 5^2 = 50
B represents 3^2 = 9
9 * 50 = 450 = the number that A + B represents.

So I feel there could be some fun geometric things to do here that may
correspond wierdly to what it means for the numbers. Like If i had two
vectors what how would the cross product relate to the numbers that
those vectors represented? Or in the example of the number 27, this
number would have the same distance from zero as
2*3*5*7*11*13*17*19*23, which is wierd. The appealing part of this way
of looking at it, is that i feel more flexible looking for patterns in
vectors than i do in integer lists.

Patrick Hamlyn wrote:
 Quote: "doctormungmung@gmail.com" wrote: This is my first post here, so excuse any etiquette i accidentally miss. My method was to treat each prime as a dimension in an infinite dimensional space. So ... the number 2 would correspond to the vector [1,0,0,0 .....] the number 3 would correspond to the vector [0,1,0,0 .....] the number 5 would correspond to the vector [0,0,1,0 .....] and as all integers are primes or composites, you would be able to represent every integer in this space. Not immediately apparent what you mean here. Are you limiting yourself to unit vectors (with plus or minus sign, else you couldn't represent '4'), or are you allowing any positive integers or any reals? I'll assume you mean non-negative integers, because if you allow reals you could just write every number X as [X/2,0,0,0,0 ...] also i think the map could work with the positive rationals if 1 mapped to [0,0,0,0 ......] Not sure what you mean here either. How would you represent 2/3? Also, any system with a non-zero quantity mapped to the zero vector is probably going to break in many other ways. 0+0 equals 1+1 for example. You just broke all arithmetic in the system. anyway, the reason this seemed like a good idea, is that each point that has a distance 1 away from zero is prime... This would support my assumption that only integers can be used in each dimension, otherwise this statement would not be true. Numbers of the form P1*sqrt(1/2) + P2*sqrt(1/2) all can be represented by a vector having distance 1 from the origin, where P1, P2 are distinct primes. ... All other numbers are composite. So I was hoping to find a pattern in the distance away from zero, as we go up the number line. I made graphs on www.tonydoran.com. And it seems that there is some order to the graph. The way I like to think the distance is kind of like a measure of how prime the number is, if the number has large distance, it is less prime, and a smaller one is more prime.... (it does sound wierd). It would seem that most composites have more than one representation in this system, as indeed do most primes. Even if you specify that the only representation for any number is the one that is closest to zero, some numbers have several representations equally close to zero. In fact it's not obvious to me how to easily find the 'closest to zero' representation for very large numbers. What I am also curious is to find ways to describe how this space gets populated. Also i am interested about finding other metrics that are easy to calculate in many dimensions. 'Dimensionality' could be one. Primes require one dimension to describe, all composites can be represented by two dimensions (usually not the representation closest to zero). If we limit the domain of allowable values for each dimension, some composites might be representable only by using three or more dimensions. For some such domain (eg +/- 1) it might be interesting to define the sequence of numbers N1,N2,N3... being 'the smallest number which has dimensionality n'. Alternately, using the domain 'positive integers' for each dimension the 'closest to zero' representation might require three or more dimensions. The 'minimum dimension sequence' for such a system would start (did this in my head so errors & omissions excepted) (2,8,27,...) In other words '2' is the smallest number which can be described minimally with a single dimension, 8 is the smallest number which has a minimum representation which has two dimensions [0,1,1] (we use this as shorthand for [0,1,1,0,0,0,...]) and I think 27 is the smallest which requires three dimensions, eg [0,1,1,0,0,0,0,1] or [0,1,0,1,0,0,1] or [0,1,0,0,1,1] (ie 3+5+19 or 3+7+17 or 3+11+13). After that it gets tricky, since you will eventually have to look at vectors involving digits larger than 1. In fact 4 and 6 are such a numbers, I think their minimum representations are [2] and [0,2] respectively. It might even be interesting to define 'each dimension has a domain such that every number has a unique minimum representation in the system'. Not sure whether this works though. Clarify the rules a bit, so I know I'm on the right track... I looked at your site, and you appear to have not considered the 'minimum distance from zero' representation, eg 27 you represent as [0,7].
moriman
science forum beginner

Joined: 06 Apr 2006
Posts: 26

Posted: Fri Jul 07, 2006 6:38 pm    Post subject: Re: My prime research.

<doctormungmung@gmail.com> wrote in message
 Quote: let me clarify the way i thought this should work each integer can be written as : p(1)^a * p(2)^b * p(3)^c ...... where p(n) is the n-th prime i want to represent each number as vector whose values would be: [a,b,c.........] so in the case of the number 6: 2^1 * 3^1 * 5^0 *7^0 ..... [1,1,0,0,.......] or in the case of the number 27: 2^0 * 3^3 * 5^0 * 7^0 ...... [0,3,0,......] or in the case of the number 2450: 2^1 * 3^0 * 5^2 * 7^2 * 11^0 ...... [1,0,2,2,0,0,......] this would imply the 1 = [0,0,0,0,.....]

Although I am doubtful if your idea will lead anywhere meaningful, would not
a better approach be to have

[0,0,0,0.......] = 0
[1,0,0,0.......] = 1
[0,1,0,0.......] = 2
[0,3,4,2.......] = 16200 (2^3 * 3^4 * 5^2)

mori
Proginoskes
science forum Guru

Joined: 29 Apr 2005
Posts: 2593

Posted: Fri Jul 07, 2006 10:41 pm    Post subject: Re: My prime research.

moriman wrote:
 Quote: doctormungmung@gmail.com> wrote in message news:1152280578.262824.161630@s13g2000cwa.googlegroups.com... let me clarify the way i thought this should work each integer can be written as : p(1)^a * p(2)^b * p(3)^c ...... where p(n) is the n-th prime i want to represent each number as vector whose values would be: [a,b,c.........] so in the case of the number 6: 2^1 * 3^1 * 5^0 *7^0 ..... [1,1,0,0,.......] or in the case of the number 27: 2^0 * 3^3 * 5^0 * 7^0 ...... [0,3,0,......] or in the case of the number 2450: 2^1 * 3^0 * 5^2 * 7^2 * 11^0 ...... [1,0,2,2,0,0,......] this would imply the 1 = [0,0,0,0,.....] Although I am doubtful if your idea will lead anywhere meaningful, would not a better approach be to have [0,0,0,0.......] = 0 [1,0,0,0.......] = 1 [0,1,0,0.......] = 2 [0,3,4,2.......] = 16200 (2^3 * 3^4 * 5^2)

The problem is that (the vector representing 1) + (the vector
representing 2) is not the same as (the vector representing 1*2).
(Think logarithms.)

0 would be [-infinity, -infinity, -infinity, ...]

--- Christopher Heckman
Proginoskes
science forum Guru

Joined: 29 Apr 2005
Posts: 2593

Posted: Fri Jul 07, 2006 10:48 pm    Post subject: Re: My prime research.

doctormungmung@gmail.com wrote:

I suppose doctorhungmung@gmail.com was taken ... 8-)

 Quote: let me clarify the way i thought this should work each integer can be written as : p(1)^a * p(2)^b * p(3)^c ...... where p(n) is the n-th prime i want to represent each number as vector whose values would be: [a,b,c.........] so in the case of the number 6: 2^1 * 3^1 * 5^0 *7^0 ..... [1,1,0,0,.......] or in the case of the number 27: 2^0 * 3^3 * 5^0 * 7^0 ...... [0,3,0,......] or in the case of the number 2450: 2^1 * 3^0 * 5^2 * 7^2 * 11^0 ...... [1,0,2,2,0,0,......] this would imply the 1 = [0,0,0,0,.....] and 2/3 = [1,-1,0,0,0,0......] Irrationals are not going to make sense here because they are not going to have multiple representations

Because they _are_ going to have multiple representations.

 Quote: Adding vectors in this system is equivalent to multiplying the numbers that they represent. meaning that if you had A = [1,0,2,0......] B = [0,2,0,0......] then A + B = [1,2,2,0.....] A represents 2^1 * 5^2 = 50 B represents 3^2 = 9 9 * 50 = 450 = the number that A + B represents.

Like logarithms. Also, multiplying a vector representing m by an
integer n results in the vector for m^n. And the p-adic norm of a
vector representing r (a rational) is just the entry in p's position.

 Quote: So I feel there could be some fun geometric things to do here that may correspond wierdly to what it means for the numbers. Like If i had two vectors what how would the cross product relate to the numbers that those vectors represented?

You only have a cross-product in R^3, so the question is meaningless.
You could try looking at the dot product of two vectors, though.

 Quote: Or in the example of the number 27, this number would have the same distance from zero as 2*3*5*7*11*13*17*19*23, which is weird.

You have more than one norm available here, which opens up other
possibilities.

http://mathworld.wolfram.com/VectorNorm.html

I'd think that you should only really be able to talk about the 1-norm
and the infinity-norm, since the others can result in irrational
numbers.

 Quote: The appealing part of this way of looking at it, is that i feel more flexible looking for patterns in vectors than i do in integer lists.

Keep going, and tell us what you find out!

--- Christopher Heckman

 Quote: Patrick Hamlyn wrote: "doctormungmung@gmail.com" wrote: This is my first post here, so excuse any etiquette i accidentally miss. My method was to treat each prime as a dimension in an infinite dimensional space. So ... the number 2 would correspond to the vector [1,0,0,0 .....] the number 3 would correspond to the vector [0,1,0,0 .....] the number 5 would correspond to the vector [0,0,1,0 .....] and as all integers are primes or composites, you would be able to represent every integer in this space. Not immediately apparent what you mean here. Are you limiting yourself to unit vectors (with plus or minus sign, else you couldn't represent '4'), or are you allowing any positive integers or any reals? I'll assume you mean non-negative integers, because if you allow reals you could just write every number X as [X/2,0,0,0,0 ...] also i think the map could work with the positive rationals if 1 mapped to [0,0,0,0 ......] Not sure what you mean here either. How would you represent 2/3? Also, any system with a non-zero quantity mapped to the zero vector is probably going to break in many other ways. 0+0 equals 1+1 for example. You just broke all arithmetic in the system. anyway, the reason this seemed like a good idea, is that each point that has a distance 1 away from zero is prime... This would support my assumption that only integers can be used in each dimension, otherwise this statement would not be true. Numbers of the form P1*sqrt(1/2) + P2*sqrt(1/2) all can be represented by a vector having distance 1 from the origin, where P1, P2 are distinct primes. ... All other numbers are composite. So I was hoping to find a pattern in the distance away from zero, as we go up the number line. I made graphs on www.tonydoran.com. And it seems that there is some order to the graph. The way I like to think the distance is kind of like a measure of how prime the number is, if the number has large distance, it is less prime, and a smaller one is more prime.... (it does sound wierd). It would seem that most composites have more than one representation in this system, as indeed do most primes. Even if you specify that the only representation for any number is the one that is closest to zero, some numbers have several representations equally close to zero. In fact it's not obvious to me how to easily find the 'closest to zero' representation for very large numbers. What I am also curious is to find ways to describe how this space gets populated. Also i am interested about finding other metrics that are easy to calculate in many dimensions. 'Dimensionality' could be one. Primes require one dimension to describe, all composites can be represented by two dimensions (usually not the representation closest to zero). If we limit the domain of allowable values for each dimension, some composites might be representable only by using three or more dimensions. For some such domain (eg +/- 1) it might be interesting to define the sequence of numbers N1,N2,N3... being 'the smallest number which has dimensionality n'. Alternately, using the domain 'positive integers' for each dimension the 'closest to zero' representation might require three or more dimensions. The 'minimum dimension sequence' for such a system would start (did this in my head so errors & omissions excepted) (2,8,27,...) In other words '2' is the smallest number which can be described minimally with a single dimension, 8 is the smallest number which has a minimum representation which has two dimensions [0,1,1] (we use this as shorthand for [0,1,1,0,0,0,...]) and I think 27 is the smallest which requires three dimensions, eg [0,1,1,0,0,0,0,1] or [0,1,0,1,0,0,1] or [0,1,0,0,1,1] (ie 3+5+19 or 3+7+17 or 3+11+13). After that it gets tricky, since you will eventually have to look at vectors involving digits larger than 1. In fact 4 and 6 are such a numbers, I think their minimum representations are [2] and [0,2] respectively. It might even be interesting to define 'each dimension has a domain such that every number has a unique minimum representation in the system'. Not sure whether this works though. Clarify the rules a bit, so I know I'm on the right track... I looked at your site, and you appear to have not considered the 'minimum distance from zero' representation, eg 27 you represent as [0,7].
Frederick Williams

Joined: 19 Nov 2005
Posts: 97

Posted: Sat Jul 08, 2006 1:26 am    Post subject: Re: My prime research.

Proginoskes wrote:

 Quote: You only have a cross-product in R^3,

And R and R^7.

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jasen
science forum beginner

Joined: 28 Jun 2006
Posts: 16

Posted: Sat Jul 08, 2006 11:23 am    Post subject: Re: My prime research.

On 2006-07-07, Frederick Williams <Frederick.Williams1@antispamtesco.net.invalid> wrote:

 Quote: the number 2 would correspond to the vector [1,0,0,0 .....] the number 3 would correspond to the vector [0,1,0,0 .....] the number 5 would correspond to the vector [0,0,1,0 .....] and as all integers are primes or composites, you would be able to represent every integer in this space. Represented how? I mean, consider 90 = 2 3^2 5, how does that get represented in terms of [1,0,0,0,...], [0,1,0,0 ...] and [0,0,1,0 ...]?

[1,2,1,0 ... ]

 Quote: Also, if by integers you mean ...,-2,-1,0,1,2,..., you need to represent negation as well.

ooh!

Bye.
Jasen
jasen
science forum beginner

Joined: 28 Jun 2006
Posts: 16

Posted: Sat Jul 08, 2006 11:33 am    Post subject: Re: My prime research.

On 2006-07-07, Patrick Hamlyn <path@multipro.N_OcomSP_AM.au> wrote:

 Quote: also i think the map could work with the positive rationals if 1 mapped to [0,0,0,0 ......] Not sure what you mean here either. How would you represent 2/3? Also, any system with a non-zero quantity mapped to the zero vector is probably going to break in many other ways. 0+0 equals 1+1 for example. You just broke all arithmetic in the system.

These vectors are analogous to logarythms of the numbers they represent,
adding two vectors results in the vector for the product og the two original
numbers.

 Quote: anyway, the reason this seemed like a good idea, is that each point that has a distance 1 away from zero is prime... It would seem that most composites have more than one representation in this system, as indeed do most primes. Even if you specify that the only representation for any number is the one that is closest to zero, some numbers have several representations equally close to zero. In fact it's not obvious to me how to easily find the 'closest to zero' representation for very large numbers.

I think the poster's idea was to represent natural numbers as vectors
of the radices of their prime factorisation. an Idea I've encountered
before - not sure where.

Bye.
Jasen
jasen
science forum beginner

Joined: 28 Jun 2006
Posts: 16

Posted: Sat Jul 08, 2006 11:38 am    Post subject: Re: My prime research.

On 2006-07-07, doctormungmung@gmail.com <doctormungmung@gmail.com> wrote:

 Quote: So I feel there could be some fun geometric things to do here that may correspond wierdly to what it means for the numbers. Like If i had two vectors what how would the cross product relate to the numbers that those vectors represented?

I don't think cross product is defined for pairs of vectors unless they are
of order 3.

dot product would be non-zero where numbers share a common factor

Bye.
Jasen
moriman
science forum beginner

Joined: 06 Apr 2006
Posts: 26

Posted: Sat Jul 08, 2006 3:00 pm    Post subject: Re: My prime research.

"Proginoskes" <CCHeckman@gmail.com> wrote in message
 Quote: moriman wrote: doctormungmung@gmail.com> wrote in message news:1152280578.262824.161630@s13g2000cwa.googlegroups.com... let me clarify the way i thought this should work each integer can be written as : p(1)^a * p(2)^b * p(3)^c ...... where p(n) is the n-th prime i want to represent each number as vector whose values would be: [a,b,c.........] so in the case of the number 6: 2^1 * 3^1 * 5^0 *7^0 ..... [1,1,0,0,.......] or in the case of the number 27: 2^0 * 3^3 * 5^0 * 7^0 ...... [0,3,0,......] or in the case of the number 2450: 2^1 * 3^0 * 5^2 * 7^2 * 11^0 ...... [1,0,2,2,0,0,......] this would imply the 1 = [0,0,0,0,.....] Although I am doubtful if your idea will lead anywhere meaningful, would not a better approach be to have [0,0,0,0.......] = 0 [1,0,0,0.......] = 1 [0,1,0,0.......] = 2 [0,3,4,2.......] = 16200 (2^3 * 3^4 * 5^2) The problem is that (the vector representing 1) + (the vector representing 2) is not the same as (the vector representing 1*2). (Think logarithms.) 0 would be [-infinity, -infinity, -infinity, ...]

Yup, I wasn't really thinking from the vector angle so I stand corrected
A bit dangerous though, for the OP to ignore zero.

mori

 Quote: --- Christopher Heckman
Felicis@gmail.com
science forum beginner

Joined: 19 Jul 2006
Posts: 4

Posted: Thu Jul 20, 2006 7:13 am    Post subject: Re: My prime research.

Hi there-

 Quote: I think the poster's idea was to represent natural numbers as vectors of the radices of their prime factorisation. an Idea I've encountered before - not sure where.

I recently came upon this same idea on:

http://mathworld.wolfram.com/ExponentVector.html

from the discussion about Dixon's factorization method. I was
interested in the idea of an exponent vector as a friend had thought up
the idea while we were in college and I hadn't thought it would be of
any use! (Of course- he came up with it independently, and long after
it had already been used... Oh well, still, it was a good excuse to
call him up and chat about old times...)

Anyways- perhaps that is where you saw it?

cheers-
Eric
Felicis@gmail.com
science forum beginner

Joined: 19 Jul 2006
Posts: 4

Posted: Thu Jul 20, 2006 2:26 pm    Post subject: Re: My prime research.

 Quote: What I am also curious is to find ways to describe how this space gets populated. Also i am interested about finding other metrics that are easy to calculate in many dimensions. angle would be one, may not be very intersting in this case though.

I don't know- clearly a pair of numbers relatively prime would be
perpendicular, and any smaller angle would be related somehow to their
common factors...

Isn't there already a body of work on infinite dimensional spaces?
Would that be applicable here?

cheers-
Eric

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