A Measure of How un-prime a Number is?

[Newtime]

Note: I have never formally studied number theory so what I'm about to ask may be either completely trivial or completely meaningless. In either case, I don't know what to search for to find my answer since I don't know the terminology.

Is there a measure of how far from a prime a given natural number is? For example, I was thinking a good measure might be, given the prime decomposition, how many unique factors there are. So we might say $$100=2^2 \cdot 5^2$$ is closer to being prime than say $$150=5^2 \cdot 3 \cdot 2$$. However, the first problem I see here is that it might be the case that some number with a few hundred digits with 3 distinct prime factors is considered as un-prime as the number $$30=2\cdot 3 \cdot 5$$. The reason I think this may be considered a problem is that one would think that as the number grows in size (number of digits), the number of factors is more likely (in some sense) to have more prime factors in its decomposition whereas the number 30 stands no chance of having a large number of prime factors in its decomposition. So perhaps using this method one would need to normalize things and define some sort of index.

Has all this been done? If so, can someone direct me to an expository article on the subject? If not, I would assumed it's because this leads nowhere since it seems to be a natural question to ask. If it leads nowhere, why?

 

[micromass]

Hi Newtime

Note: everything that followed here is made up by me. Maybe it already exists, but I don't know

There might be some solutions to your problem:

• Let d denote the divisor function, that is d(n) is the number of divisors of n. So, example d(2)=2, d(3)=2, d(4)=3,... See http://en.wikipedia.org/wiki/Divisor_function , I'm talking about $$\sigma_0$$.
  I'll denote dim(n) for the measure of how unprime a number is. I do this, because I intuitively see this as a dimension. The formula you seem to wanting is:
  
  $$dim(n)=\frac{d(n)-2}{n}$$
  
  Every prime number now has dimension 0. And generally, how bigger the number, how lower the dimension.
• A variation of above that gives more weight to the largeness of the number n. Consider s>1 fixed, then define
  
  $$dim(n)=\frac{d(n)-2}{n^s}$$
  
  As seen from the article http://en.wikipedia.org/wiki/Divisor_function , this has a lot of connections to the Riemann-zeta function. Specifically, we have
  
  $$\sum_{n=1}^{+\infty}{dim(n)}=\zeta(s)^2+2\zeta(s)$$
  
• Another variation of the above might be to change d for $$\sigma_n$$, the n'th-divisor function.
  
• An entire different direction may be obtained by dimensions of rings. Specifically, we can define dim(n) as the Krull-dimension of $$\mathbb{Z}_n$$ (or some other dimensions such as Goldie dimension, homological dimension, projective dimension). All of these things give us good definitions of dimensions. Check out an abstract algebra course for this.
  
• We can also view the subgroup lattice of the group $$\mathbb{Z}_n$$. There are some nice numerical invariants that may come in handy, like height and width of the lattice. Since these lattices are distributive and hence semimodular, the height function satisfies good notions of dimensions. Check out an abstract algebra and lattice theory text for this.
  
• We can also let dim(n) be the number of groups of order n. Every prime number will then have dimension 1. And how bigger n, how bigger the dimension in general. Check out an abstract algebra text for this.

I hope I gave you some ideas on how to proceed...

Since you said to have no training in mathematics, I realize I probably spoke over your head here. But I hope to have given you some motivation and ideas that you can research. If you do this, then you'll discover lots of beautiful mathematics + an answer to your question...

 

[Newtime]

Thanks for the reply, micromass.

The $$dim(n)$$ seems very close to what I want. It's not as directly related to the prime decomposition but I like how it takes into account the "size" of a number.

I've had several courses in algebra but have not studied the dimension of rings. Is this a topic usually introduced in a course on commutative algebra?

Personally, since I'm interested in group theory, I think your thoughts on the subgroup lattice of $$\mathbb{Z}_n$$ are the most interesting. The only potential problem here is that, although still elementary, it may take more time to find this lattice than it would to find the prime decomposition of a number or the value of its divisor function.

 

[micromass]

Thanks for the reply, micromass.

The $$dim(n)$$ seems very close to what I want. It's not as directly related to the prime decomposition but I like how it takes into account the "size" of a number.

Well, the number of divisors of a function is closely related to the prime decomposition, but you might also consider doing

$$dim(n)=\frac{\text{number of primes in the decomposition}}{n}$$

The reason why I did not do that is because number theorists like to work more with divisors than with the number of primes in the decomposition. Also, I find the relationship with the Riemann-Zeta function elegant.

I've had several courses in algebra but have not studied the dimension of rings. Is this a topic usually introduced in a course on commutative algebra?

Yes, this would be commutative algebra. It is a really important topic in algebraic geometry as the dimension of a ring can be used to describe the dimension of a "geometric space" (= scheme).
The book by Eisenbud: "Commutative algebra with a view on algebraic geometry" describes some of these things.

Also, with every number, we can associate $$\mathbb{Z}_n$$ a ring. And with this ring, we can associate a scheme (which is some kind of geometric space). The more complicated n is, the more complicated the scheme. So I thought it would make sense to use scheme invariants like dimension to describe n.

Personally, since I'm interested in group theory, I think your thoughts on the subgroup lattice of $$\mathbb{Z}_n$$ are the most interesting. The only potential problem here is that, although still elementary, it may take more time to find this lattice than it would to find the prime decomposition of a number or the value of its divisor function.

Hmm, this is an interesting problem. Check out this link: http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n2/paper4/v7n2-4pd.pdf

There, they try to describe some easy cases of $$\mathbb{Z}_n$$.

 

[Newtime]

Well, the number of divisors of a function is closely related to the prime decomposition, but you might also consider doing

$$dim(n)=\frac{\text{number of primes in the decomposition}}{n}$$

The reason why I did not do that is because number theorists like to work more with divisors than with the number of primes in the decomposition. Also, I find the relationship with the Riemann-Zeta function elegant.

I definitely agree with you there.

Yes, this would be commutative algebra. It is a really important topic in algebraic geometry as the dimension of a ring can be used to describe the dimension of a "geometric space" (= scheme).
The book by Eisenbud: "Commutative algebra with a view on algebraic geometry" describes some of these things.

Also, with every number, we can associate $$\mathbb{Z}_n$$ a ring. And with this ring, we can associate a scheme (which is some kind of geometric space). The more complicated n is, the more complicated the scheme. So I thought it would make sense to use scheme invariants like dimension to describe n.

I'm going to be using Eisnebud's book this fall in a course on commutative algebra. Hopefully in a few months I can better understand some of what you've said here but I like the general idea and it seems like this too could be the measure I'm looking for.

Hmm, this is an interesting problem. Check out this link: http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n2/paper4/v7n2-4pd.pdf

There, they try to describe some easy cases of $$\mathbb{Z}_n$$.

Interesting; thanks for the link. Also, the program GAP could be of great help here since I'm sure there's a function built in that computes these lattices for finite cyclic groups. Of course, with respect to the entire natural number line, there are a lot "more" large numbers than small so this approach's efficiency may taper off.

 

[micromass]

Well, if you use GAP, then you can very easily calculate the Krull dimension or the subgroup lattice. If you're on linux, then you might consider the add-on XGAP which can actually display the subgroup lattice.

However, I find GAP a bit lacking in working with lattices. Sure, it can calculate the subgroup lattice, but that's it basically. It can't even check for distributivity as far as I'm aware. The program Sage is much better in that respect...

 

[Eynstone]

You might find S. Ramanujan's paper on Highly COmposite Numbers interesting.

 

[Newtime]

Well, if you use GAP, then you can very easily calculate the Krull dimension or the subgroup lattice. If you're on linux, then you might consider the add-on XGAP which can actually display the subgroup lattice.

However, I find GAP a bit lacking in working with lattices. Sure, it can calculate the subgroup lattice, but that's it basically. It can't even check for distributivity as far as I'm aware. The program Sage is much better in that respect...

I haven't worked with GAP extensively so I can't really comment on its strengths/weaknesses but I (naively) thought it was the only such program. I'll have to look into Sage, thanks.

You might find S. Ramanujan's paper on Highly COmposite Numbers interesting.

Thanks for the suggestion. It looks like this could be the answer I was looking for.