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

[Hurkyl]

As for Hurkyl, I still think he ought to re-do his 'test' without being rather obtuse towards my thesis.

The problem, as shmoe pointed out, is that your postings are vague; we "can't" test what you say, because you don't say anything testable. I'm left to make my best guess as to what you mean, and apparently I tend to be wrong about that.


I've streamlined my code, so I could look for even more interesting examples. The latest is this:

6412372842 = 2 * 3 * 1068728807
6412372843 = 6412372843
6412372844 = 2^2 * 1603093211

This is as minimal as an example can get -- among the neighbors of a prime greater than 3, one must be divisible by four, and the other must be divisible by two. And, of course, one must be divisible by 3.

This is a moderately sized example where what's "left over" is prime.

I can find more of this size... but I'd have to devise a more clever approach than brute force to find examples much bigger than that. (A sieve, maybe?)


I conjecture that you can find arbitrarily large examples like this: primes P such that P - 1 and P + 1 are 2 * 3 * Q and 4 * R (not necessarily in that order) where Q and R are both primes.


The numbers involved, there, don't have unusally many factors. In fact, on average, numbers in that range have 3.85 prime factors. (3.08 distinct prime factors)

 

[marteinson]

Thanks, Shmoe, that's what this kind of forum is for. But do you mean, by "this is enough to get my vote" "to move this to TD" that the moderators should feel free to move anything to a location that has no relevance to its discussion, because of an unstated and anonymous judgment they form regarding the quality of the posts? If so, we could envision a physicsforum where things are variously located where they should be, or somewhere far from the correct location, according to private opinions and secret desires to keep things from being seen and read. This would mean contributors have had their freedoms limited.

I'll look into the work you suggest (if I can find them, "you should learn to" cite academic works properly, to use your own type of language).

Also, I don't think the site is meant to provide a forum to mock other contributors. I see it as a place to share ideas and learn. Real teachers wouldn't subject any contributors to derision, I am sure.

 

[marteinson]

Many things I said were perfectly clear, to anyone who wasn't aiming at distorting them for the purpose of posing as a gadfly. The literature still has all sorts of claims that there is no explanation for the distribution of the primes, and that we lack any reason why primes often occur in pairs. That is why I decided to post my ideas here, in case they interested others. I apologize if I abused my contributing privileges.

Also, I think the following is 'testable", though I never suggested Hurkyl test it (especially not with the procedure of saying 'he says prims are highly divisible numbers, I'll prove he's wrong by defining these as multiples of 210'): if n has the factors p, q, r, s, t...v, then n-1 and n+1 will lack these factors; it is the distribution of factors at n and specific locations near n, such as n+2 and n-2, that makes n-1 and n+1 poor in factors or even prime.

To me, that is an interesting property of the natural numbers, and worth mentioning and discussing in a post on number theory.

 

[shmoe]

I'll look into the work you suggest (if I can find them, "you should learn to" cite academic works properly, to use your own type of language).

I didn't bother with specific links or references because these are "classical results" that have been around about 100 years and you can find information or references in most basic number theory texts (or even google). Several times you've mentioned what "the literature" has to say, I suppose I jumped to the conclusion that you had some familiarity with it. There's so many books to suggest, but maybe you could give Hardy and Wrights "An introduction to the theory of numbers" a whirl. (the wheel sieve isn't so common or old, but Pritchards paper used to be on Bernstein's website but I can't seem to find it there anymore).

You might see mathematicians say that not much is understood about primes, but this is really just a comparison with what they wished they knew. I mean the power of the prime number theorem with even the currently known "weak" (compared to what's hoped to be true) error term is just impressive. You might find Hardy and Littlewoods asymptotic conjectures about twin primes suprisingly accurate if you do some computations (though of course these numeric computations prove nothing). Recently (~30? years) it was shown (Chen?) there are infinitely many primes p where p+2 is either prime or has at most two prime factors. So even though no one has proven there are infinitely many prime pairs, it's not like nothing is known in this direction or that we're just bumbling around in the dark.

As for mockery, I call it as I see it.You continually avoid any precise definitions and I really don't understand why. You make up terms like "poor in factors" or "highly divisible" and expect people to understand. You either have to have some way of classifying what "poor in factors" means or you're not talking about anything that can be analyzed in a meaningful way. This is why I think TD is where this post belongs, it's "pseudomathematics" until you remove this cloud of ambiguity. I don't know what to suggest other than read some more mathematics to get a better feel for how they communicate (e.g. Hardy & Wright) or take a number theory course near you.

 

[marteinson]

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. For example, 24 has many factors (2*12, 3*8, 4*6) while 25 has few (5*5) and is therefore poorer in factors. 23 has none at all (other than the trivial pair, 1 and itself) and is prime as a result of this lack, which I prefer to view as a result of the way in which the factors available to numbers in the vicinity of 24 (such as 2, 3, 4, 5, 11 but unlike 13, 17 and larger primes) are all distributed to other numbers.

If Hurkyl's main objection is that the concentration of factors not only n, but n+-2, n+-3, n+-4 and the other composites near n normally also contribute to the way in which n+-1 is left out of the factor distribution when n+-1 are prime, I agree with that, especially in larger and larger n.

But I don't think what I have been saying is ambiguous, just not as clear as mathematical symbols (I don't have Latex). If people are interested in reading it, and would like to know more clearly what I mean, they could always ask, before writing long and arduous objections.

The observation I have put forth here is that a dearth of factors occurs adjacent to a wealth of factors on the natural number set, and that primes, being the most extreme dearth of factors, occur next to (one above and/or one below) numbers having a wealth of factors, relative to the other numbers in the vicinity of n.

To me it was natural to use the language in the way I did, and didn't intend to "continually avoid" precise definitions, I'm doing my level best, really, not to be imprecise.

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?

Thanks for the recommended reading, I'll read it with interest.

 

[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'.