What is the relationship between primes and the distribution of factors?

[marteinson]

Professor Coombes at MIT also uses the term "highly divisible number", so I didn't make it up and evidently MIT faculty and students don't find it hard to understand. That's why I said some contributors here were being obtuse, i.e. deliberately using the claim of incomprehensibility as a polemical artifice.

Here's where he uses it, and on that page, he is also describing the manner in which adding one to a highly divisible number gives a highly un-divisible number. I don't think it becomes pseudo-mathematics when I say it just because I'm not an MIT professor, does it?

http://odin.mdacc.tmc.edu/~krc/numbers/infinite.html

Anyhow, I'll now try and look into the readings Shmoe kindly points out, to see if what I have been saying is wrong, as Hurkyl has maintained, or contains nothing new, as Shmoe seemed to imply.

 

[marteinson]

Okay I've done some of my homework as Shmoe suggested.

Highly divisible numbers, also called highly composite numbers, were considered to have been discovered or developed by Ramanujan.

Ramanujan, S.: Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 87.

 

[shmoe]

Professor Coombes at MIT also uses the term "highly divisible number", so I didn't make it up and evidently MIT faculty and students don't find it hard to understand. That's why I said some contributors here were being obtuse, i.e. deliberately using the claim of incomprehensibility as a polemical artifice.

Here's where he uses it, and on that page, he is also describing the manner in which adding one to a highly divisible number gives a highly un-divisible number. I don't think it becomes pseudo-mathematics when I say it just because I'm not an MIT professor, does it?

No it's not pseudo mathematics, but notice that any precise meaning of the terms are not at all needed for what he's doing. He's giving an informal description of how one might go about finding or understanding this proof (he even says as much) and you could remove those terms without any problems. You'll notice in his http://odin.mdacc.tmc.edu/~krc/numbers/infitude.html that he has solid definitions for everything (note the links to all needed definitions and earlier propositions) and these vague terms are absent.

To make things clear, "poor in factors" means a number has few of them (not many). Highly divisible means a number can be divided in many distinct ways. The words "few" and "many" are relative.

This is exactly the kind of vagueness that's a problem. Relative to what? Some other numbers, but which? How close? Is there some cut-off between few and many or is there some grey area?

What would you, who have read this thread, call a number whose number of factors represents a local maximum on the relation between n and its number of factorizations?

You would have to define what you mean by "locally" first, what counts as local?

There are the standard # of divisors function and the sum of divisors functions (notations will vary). There are notions of degrees of divisibility that are relative only to a number itself, I've seen the terms abundant, deficient and perfect all relating the sum of divisors back to the number. There are many more specialized terms, but I can't think of anything offhand that's aiming for any kind of local property.

Highly divisible numbers, also called highly composite numbers, were considered to have been discovered or developed by Ramanujan.

Ahh yes. Highly composite rings a bell, but I don't think I've seen highly divisible before (I may have though!). Note that it doesn't seem to be what you're describing and not what Coombes is talking about either. It's not really a definition that comes up often (one of those "specialized" terms), which would explain his informal use of the words not referring to Ramanujan's definition.

 

[WaterBreath]

marteinson - I think I see why they are saying that this fits in "Theory Development", though I don't think it really fits under a "Physics" category. If there's a general understanding that a topic such as "Number Theory" will be reserved for talking about known theorems, formalized theories, the history of the field, etc., then "new theories", which was kind of implied in your original posts, wouldn't really belong there. People looking for help or clarification might be confused as to the formal status of the ideas.

You might want to look into "Explaining the wheel sieve" by Pritchard. It puts this into practice in a more sophisticated way to generate a list of primes. It's also essentially using your recursive idea, and using larger "wheels" at each stage (the paper has more details).

Thanks shmoe, I will definite look into this. Very encouraging to see that some ideas I've had on my own actually might lead somewhere worthwhile!

I'm not entirely sure I see his opinion that the definition is recursive in the same way he does

That's probably because, as far as I can tell, you and I are considering different things to be "fundamental". You're considering the distribution of prime multiples to be "fundamental", whereas I'm saying that the locations of those earlier primes are at a lower level yet than that. In short, the multiples are where they are because of where the earlier primes are.

If primes are, as I've read many places, "the building blocks of numbers", then they would essentially be the "elements" of the integers. A scattering of indivisible numbers can be used to build everything that falls in between. In this respect, I consider them fundamental. But what is it that determines whether a given large number k is prime or not? As you've correctly noted, it is the fact that of all the prime numbers p where $p < \sqrt{k}$, there is no integer m such that $k=pm$. Rather than the multiples falling at k, they fall at numbers near k. By which of course I mean the nearest multiples pm satisfy $pm >= k - p, pm <= k + p, pm \neq k$. Put another way, if k is to be prime, the ranges $[k - p, k)$ and $(k, k + p]$ must include multiples of all primes p where $p < \sqrt{k}$.

So the primeness of a large number depends on the locations of multiples of smaller primes. And the locations of those primes depends on the location of the multiples of yet smaller primes, until you get down to the most fundamental of all primes: 2, which is prime only because there are no integers between it and unity which could even be candidates for being factors.

So the set of primes is recursive, or iterative, in that you can set the base condition for an algorithm to generate the set of the first n primes (call it P(n) ), and from that build all the rest after it. All you need to know is that P(1) is 2, and then P(n) where n > 2 is just a couple of GCD and LCM calculations, and some set differences, using the set given by P(n-1). But those functions themselves are iterative, so it's not a very efficient algorithm. Lots of loops and recursion. Yes, most things in mathematics are, at fundamental levels of definition, recursive. But there are also many many "shortcuts". I'm sure there are better algorithms than the one I vaguely described, but I've never heard of any algorithms that are proven to generate the nth prime that are not also recursive or iterative in some way. Am I wrong on that, anyone? I'd be greatly interested to hear of one that's not.

 

[marteinson]

Thanks to both recent posters for their interesting input. I'll think about all these ideas too. Still, I think it useful to consider prime distribution in terms of factor distribution and the interference (prim nodes and prime antinodes that occur so near to the prim nodes) between the factor wheels' wave-like phenomena and their 'interferogram'.