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Message |
doctormungmung@gmail.com
science forum beginner
Joined: 06 Jul 2006
Posts: 2
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 Posted: Thu Jul 06, 2006 6:10 pm Post subject:
My prime research.
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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.
also i think the map could work with the positive rationals if 1 mapped
to [0,0,0,0 ......]
anyway, the reason this seemed like a good idea, is that each point
that has a distance 1 away from zero is prime. 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).
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. |
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Frederick Williams
science forum addict
Joined: 19 Nov 2005
Posts: 97
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| Posted: Fri Jul 07, 2006 1:20 am Post subject:
Re: My prime research.
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"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.
--
Remove "antispam" and ".invalid" for e-mail address. |
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William Elliot
science forum Guru
Joined: 24 Mar 2005
Posts: 1906
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| Posted: Fri Jul 07, 2006 1:39 am Post subject:
Re: My prime research.
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On Thu, 6 Jul 2006, doctormungmung@gmail.com wrote:
| Quote: | This is my first post here, so excuse any etiquette i accidentally
miss.
|
http://oakroadsystems.com/genl/unice.htm |
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Patrick Hamlyn
science forum beginner
Joined: 03 May 2005
Posts: 45
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| Posted: Fri Jul 07, 2006 3:05 am Post subject:
Re: My prime research.
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"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]. |
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jasen
science forum beginner
Joined: 28 Jun 2006
Posts: 16
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| Posted: Fri Jul 07, 2006 9:25 am Post subject:
Re: My prime research.
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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 |
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doctormungmung@gmail.com
science forum beginner
Joined: 06 Jul 2006
Posts: 2
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| Posted: Fri Jul 07, 2006 1:56 pm Post subject:
Re: My prime research.
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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.
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.
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" <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]. |
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moriman
science forum beginner
Joined: 06 Apr 2006
Posts: 26
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| Posted: Fri Jul 07, 2006 6:38 pm Post subject:
Re: My prime research.
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<doctormungmung@gmail.com> wrote in message
news:1152280578.262824.161630@s13g2000cwa.googlegroups.com...
| 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 |
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Proginoskes
science forum Guru
Joined: 29 Apr 2005
Posts: 2593
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| Posted: Fri Jul 07, 2006 10:41 pm Post subject:
Re: My prime research.
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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 |
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Proginoskes
science forum Guru
Joined: 29 Apr 2005
Posts: 2593
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| Posted: Fri Jul 07, 2006 10:48 pm Post subject:
Re: My prime research.
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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.
http://mathworld.wolfram.com/p-adicNorm.html
| 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" <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]. |
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Frederick Williams
science forum addict
Joined: 19 Nov 2005
Posts: 97
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| Posted: Sat Jul 08, 2006 1:26 am Post subject:
Re: My prime research.
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Proginoskes wrote:
| Quote: | You only have a cross-product in R^3,
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And R and R^7.
--
Remove "antispam" and ".invalid" for e-mail address. |
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jasen
science forum beginner
Joined: 28 Jun 2006
Posts: 16
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| Posted: Sat Jul 08, 2006 11:23 am Post subject:
Re: My prime research.
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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 |
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jasen
science forum beginner
Joined: 28 Jun 2006
Posts: 16
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| Posted: Sat Jul 08, 2006 11:33 am Post subject:
Re: My prime research.
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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.
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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 |
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jasen
science forum beginner
Joined: 28 Jun 2006
Posts: 16
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| Posted: Sat Jul 08, 2006 11:38 am Post subject:
Re: My prime research.
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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?
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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 |
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moriman
science forum beginner
Joined: 06 Apr 2006
Posts: 26
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| Posted: Sat Jul 08, 2006 3:00 pm Post subject:
Re: My prime research.
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"Proginoskes" <CCHeckman@gmail.com> wrote in message
news:1152312073.393780.183420@s13g2000cwa.googlegroups.com...
| 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, ...]
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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
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Felicis@gmail.com
science forum beginner
Joined: 19 Jul 2006
Posts: 4
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| Posted: Thu Jul 20, 2006 7:13 am Post subject:
Re: My prime research.
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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.
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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 |
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