Geometric representation of composite numbers

[SW VandeCarr]

Some years ago I used the device of representing composite numbers by rectangular forms to demonstrate the structure of numbers to third grade students. Primes were represented by lines of various lengths. Number 10 would be a 2x5 rectangle and 20 a 2x2x5 rectangular solid. (I used various devices for dimensions > 3). Clearly this representation assigns trigonometric properties to composite numbers which might be deserving of study. I haven't run across any formal studies of these properties, but it seems they might give some insights into the structure of the natural numbers. Can anyone direct me to any studies of these properties assuming they might be worth studying.

 

[Hurkyl]

Clearly this representation assigns trigonometric properties to composite numbers which might be deserving of study.

It's not clear to me. It's not really even clear to me what the point of this representation is, since you've thrown away pretty much all of the "integerness" of the integers.

 

[CRGreathouse]

If you express them in maximal dimension (so their expanse in each dimension is prime) they have an essentially unique representation (up to rotation and translation?). Multiplication has a simple 'additive' structure. The (dimensionless) volume of each of the representations is constant across a given number. That's the best I can do; I can't think of anything inherently geometric/trigonometric about that representation.

 

[SW VandeCarr]

It's not clear to me. It's not really even clear to me what the point of this representation is, since you've thrown away pretty much all of the "integerness" of the integers.

No. I didn't throw away the integers. I simply defined a Euclidean space where coordinates are expressed only in integral units.

I agree, I don't know where this might go. However there is a proof (Landau 1900) that if p sub k (x) is the number of integers not exceeding 'x' that have k unique prime factors, then p sub k (x) approx= (x/lnx)/(ln(lnx)^k-1)/(k-1)! My representation includes repeated prime factors to uniquely identify any composite.

 

[SW VandeCarr]

If you express them in maximal dimension (so their expanse in each dimension is prime) they have an essentially unique representation (up to rotation and translation?). Multiplication has a simple 'additive' structure. The (dimensionless) volume of each of the representations is constant across a given number. That's the best I can do; I can't think of anything inherently geometric/trigonometric about that representation.

Thanks for your reply CR. Actually the algebraic representation in terms of a vector space is more interesting. Every composite can be represented by a set of vectors whose coordinates are the permutations of its prime factors. This links composite numbers with permutation groups and rotation groups. It also allows a Pythagorean distance to be defined between composite numbers that is different than a-b or b-a on the number line.

From your reply and Hurkyl's, it doesn't appear there's any non-obscure literature on this, but I appreciate your replies. Perhaps some others here might know of any work along these lines.

 

[CRGreathouse]

However there is a proof (Landrau 1900) that if p sub k (x) is the number of integers not exceeding 'x' that have k unique prime factors, then p sub k (x) approx= (x/lnx)/(ln(lnx)^k-1)/(k-1)! My representation includes repeated prime factors to uniquely identify any composite.

If you include repeated prime factors, the result doesn't change asymptotically.

 

[CRGreathouse]

It also allows a Pythagorean distance to be defined between composite numbers that is different than a-b or b-a on the number line.

Sure, but can that pseudo-quasi-whatsit-metric do anything interesting? Actually there are several interpretations of this measure, depending on how you order the prime factors. How do you decide?

I frequently think of numbers in factored form, that is a list [2, 2, 3, 7] or matrix
[ 2 2 ]
[ 3 1 ]
[ 7 1 ]
though I don't give it geometric meaning.

 

[SW VandeCarr]

Sure, but can that pseudo-quasi-whatsit-metric do anything interesting?

I don't know. Do you? Yes, I should have specified "natural" ascending order of prime factors for a preferred distance measure between composites.

 

[CRGreathouse]

(Yes, I should have specified "natural" ascending order of prime factors for a distance measure between composites).

I'm not sure that it is natural. It was the first to come to mind, but what about d(2^2 * 3, 2 * 3)? Isn't 2 a more natural value than 3sqrt(2)? In light of d(p^k, p * p^k) = p, for example.

 

[SW VandeCarr]

I'm not sure that it is natural. It was the first to come to mind, but what about d(2^2 * 3, 2 * 3)? Isn't 2 a more natural value than 3sqrt(2)? In light of d(p^k, p * p^k) = p, for example.

If I'm the only one doing this, it's very much a work in progress. I'm open to ideas. (I seriously doubt something like this has never been published.)

For example the Pythagorean distance between 81 (0,3,3,3,3) and 80 (2,2,2,2,5) would be sqrt ((0-2)^2 + 1 + 1 +1 + (3-5)^2)) = 3.3166.. I have no idea what this metric might mean unless and until a useful or at least interesting pattern could be seen over many pairwise calculations.

 

[CRGreathouse]

I'm open to ideas.

There are lots of possibilities: something like
$$\frac{\Omega(n)\Omega(m)}{(\Omega(n)-\omega(n)+1)(\Omega(m)-\omega(m)+1)}=O(\log n\log m)$$
for the distance between m and n. (More, depending on how the factors are distributed; less, depending on how much overlap there is.)

Out of all of those possibilities, it's hard to say which would be good. Usually I'd think "all of them", but without knowing what you intend to do with them I can't really say.

 

[SW VandeCarr]

There are lots of possibilities: something like
$$\frac{\Omega(n)\Omega(m)}{(\Omega(n)-\omega(n)+1)(\Omega(m)-\omega(m)+1)}=O(\log n\log m)$$
for the distance between m and n. (More, depending on how the factors are distributed; less, depending on how much overlap there is.)

Out of all of those possibilities, it's hard to say which would be good. Usually I'd think "all of them", but without knowing what you intend to do with them I can't really say.

I edited the post you quoted, adding a straightforward example of my present approach. I'm not familiar with all number theoretical notation.
One use of omega is for volume, which in this representation is simply the product of the prime factors, ie the composite number itself. That doesn't seem to be what omega means here.

 

[CRGreathouse]

I edited the post you quoted, adding a straightforward example of my present approach.

That's the same as the one I first thought of, except that I pad with zeros to the right rather than left.

I think it's natural (as I mentioned before) to pair like primes; this doesn't affect your example because 80 and 81 are coprime. This would give numbers the property that d(m, n) = d(m/g, n/g) for g | gcd(m, n).

I'm not familiar with all number theoretical notation.
One use of omega is for volume, which in this representation is simply the product of the prime factors, ie the composite number itself. That doesn't seem to be what omega means here.

Big omega is the total number of prime factors; Omega(81) = 4. Little omega is the number of distinct prime factors; omega(81) = 1.

 

[SW VandeCarr]

Big omega is the total number of prime factors; Omega(81) = 4. Little omega is the number of distinct prime factors; omega(81) = 1.

Thanks for your input CR.

 

[SW VandeCarr]

I think it's natural (as I mentioned before) to pair like primes; this doesn't affect your example because 80 and 81 are coprime. This would give numbers the property that d(m, n) = d(m/g, n/g) for g | gcd(m, n).

Sorry to double post, but I want to respond to this. If I do this, I lose my ordering principle. For example 35 (0,5,7) and 30 (2,3,5). Should I cancel the 5's and then use natural ordering ie (0,7) and (2,3)? By following natural ordering strictly, I have consistency: sqrt ((0-2)^2 + (5-3)^2 + (7-5)^2) = 3.464..

Using your suggestion: d(35, 30)= d(35/5, 30/5) = d(7,6) so we have d((0,7),(2,3)) = sqrt ( (0-2)^2 + (7-3)^2) = sqrt 20 = 4.472.

 

[CRGreathouse]

Sorry to double post, but I want to respond to this. If I do this, I lose my ordering principle. For example 35 (0,5,7) and 30 (2,3,5). Should I cancel the 5's and then use natural ordering ie (0,7) and (2,3)? By following natural ordering strictly, I have consistency: sqrt ((0-2)^2 + (5-3)^2 + (7-5)^2) = 3.464..

My suggestion also gives consistency. It is an ordering that allows that sort of cancellation without giving two different orders.

 

[SW VandeCarr]

So this would be correct, following your suggestion (I use natural ordering after cancellation)?

 

[CRGreathouse]

Sure. I see it more as d(35, 30) = sqrt((0-2)^2 + (7-3)^2 + (5 - 5)^2) = sqrt 20, though. The cancellation is a property you can derive, not part of the definition.