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

[marteinson]

"A way of conceptualizing the nature of primes..."

We know Eratosthenes observed that the primes occur at 6n+-1. We also know that Ulam's spiral is considered interesting because it visually displays a 'striking non-random appearance' in the distribution of primes.

What strikes me, however, is that primes only occur one above or one below (frequently both above and below) the most divisible natural numbers. These are 1x2x3, or 6, and 1x2x3x4, or 24, and so on to 120, 720, etc. Notice that these factorials are the source of e, which describes the frequency of primes in 1/log(n).

I am compelled to conclude that the primes are deprived of factors by these prim numbers, which is why they only occur at the positions one above and below the multiples of six, and especially at other greater, more prim numbers (those having even more factors among the small natural numbers).

I wrote an experiemental semiotics article about this which you can see at
http://www.chass.utoronto.ca/french/as-sa/ASSA-14/article7en.html

and in it, I have constructed a modulus-6 clock-like spiral showing the regularity of the distribution of primes about positions displaced one from multiples of six. This graphic can be seen at
http://www.chass.utoronto.ca/french/as-sa/ASSA-14/modulus6-spiral90.jpg

The interesting thing about this way of thinking of prmes (that they exist because the most divisible numbers to which they are adjacent attract all the factors to themselves, depriving the primes of factors) is that it explains the reason why primes frequently occur in pairs, and the appendix to the article, an excel sheet showing the mechanical manner in which the factors are distributed (rather than the distribution of primes themselves) shows an increasing likelihood of having factors according to the series
1 + 1/1 + 1/(1x2) + 1/(1x2x3)... which sum to e, or 2.71828...

This must therefore be a way of explaining the origin of primes in terms of their positional relationship to prims, which seems to me to be further supported by the relationship between these additive probabilities and the pi function of 1/log(n) in determining the frequency, or probability, of finding a prime at n.

One more thing... this explains why there are prime pairs.

What do you think?

-- Hare-brained amateur

 

[ramsey2879]

There are only two primes that are displaced more than plus or minus one from a multiple of 6 and these are 2 and 3. This is because 2 and 3 divide 6 and 4 is a multiple of 2. This leaves only numbers equal to either 1 or 5 mod 6 that could be prime for all n > 3. Thus there is nothing remarkable about the fact that primes greater than 3 are either plus or minus 1 from a multiple of 6. The same simple explanation applies for patterns involving modulus 24 or modulus other "prims". That is numbers greater than the largest prime factor of a "prim" can't be congruent to a multiple of a prime factor of a prim mod that prim inorder for the number to itself be prime.
I agree that patterns are interesting but without a novel and logical underpinning one can't gain much insight from them. Only rarely does one observe a pattern that seems to lack a logical proof to the most skilled mathematicians and yet has no known counter example. Some conjectures may not be proven for centuries such as Fermat's last therom, others may remain without any known counter example and yet still be unproven indefinitely.
In conclusion your logic that prime pairs probably continue indefinitely due to these patterns has long been noted and is probably shared by most mathematicians but it doesn't get us any closer to an answer to the question of whether there is an infinite set of prime pairs .

 

[Hurkyl]

One thing that needs to stressed in number theory is that you really can't say you've found any sort of pattern until you look at sizes that most ordinary humans would call gigantic.

For instance, to a number theorist, 10⁹ is still considered a small number!


I don't even see the evidence for some of your assertions. 38, for example, isn't a particularly "divisible" number, yet the prime 37 appears next to it. 120 is explicitly stated as a "most divisible" number, yet 121 is not a prime.


As a demo of how easy it is to be fooled by trends observed in miniscule numbers, note that the factorials aren't the "most divisible" numbers:

12! = 2^10 * 3^5 * 5^2 * 7 * 11 has 792 factors (11*6*3*2*2)
But,
11! * 7 = 2^8 * 3^4 * 5^2 * 7^2 * 11 has 810 factors. (9*5*3*3*2)


Incidentally, if you're looking for primes, there's no reason to consider repeated prime factors in these "highly divisible numbers" -- if a number has just a single factor of, say, 2, then the numbers next to it cannot be divisible by 2. An easy way to find numbers not divisible by 2, 3, or 5 is to look for things next to multiples of 2*3*5 -- you don't have to go all the way out to 5!.

To make it more explicit, the problem with your observation is that when you use small numbers, you don't have all that many extra primes laying about that weren't involved in your product. For example, the smallest prime factor of a number less than 121 must be 7 or less. So, since 120 is divisible by 2, 3, and 5, then 119 has a 6 in 7 shot of not being divisible by 7, and thus prime. (But, as it turns out, 119 = 7 * 17, so it's not prime)

But, when you get up to, say, 12!, which is divisible by 2, 3, 5, 7, and 11, the smallest prime factor of numbers in its vicinity can be as large as 21886! I worked out by hand that a "random" number has a roughly 43% chance of being divisible by at least one of the primes from 13 through 97... maybe I'll work out by computer when I get back home what the odds are of being divisible by one of the primes from 13 through 21886.

 

[Spin_Network]

Your link here of figure one:http://www.chass.utoronto.ca/french/as-sa/ASSA-14/article7en.html

This may be of interest to your inquiry:http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm

please forgive the basic sloppy webpage...its been online for sometime and I have other things to deal with, computer webpage's are not my thing.

Now the really..really interesting thing is if one puts the base numbers into Eratosthene Sieve,

http://www.math.utah.edu/~alfeld/Eratosthenes.html

and as you know there is NO pattern that emerges, but if one place the numbers as I have in my webpage, the construction produces numbered patterns.

So if one now does this:

0.818269320][2564609051][3931005094][6279551!864][836066
1.929370431][3675710162][4042116105][7380662!975][947177
2.030481542][4786821273][5153227216][8491773!086][058288

3.141592653][5897932384][6264338327][9502884!197][169399

4.252603764][6908043495][7375449438][0613995!208][270400
5.363714875][7019154506][8486550549][1724006!319][381511
6.474825986][8120265617][9597661650][2835117!420][492622
7.585936097][9231376728][0608772761][3946228!531][503733
8.696047108][0342487839][1719883872][4057339!642][614844
9.707158219][1453598940][2820994983][5168440!753][725955

for all the numbers above and below the third set of numbers(which is obviously Pi!) then some interesting things occur!

Set out prime number linearly( I have to 40 sig numbers )..the deduct linearly the numbers above and below as I have, the above numbers are decreasing ie..3.14 above line 2.03..below line are increasing 4.25..etc..etc

This numbering system when coupled to the 'Prime Number Factor' has some other factors I have not detailed..but I am confident you will find them!

This should be a reply to original poster..!

 

[marteinson]

I don't even see the evidence for some of your assertions. 38, for example, isn't a particularly "divisible" number, yet the prime 37 appears next to it. 120 is explicitly stated as a "most divisible" number, yet 121 is not a prime.

But 37 is next to 36, a highly divisible multiple of six. Also, 121 only has a single pair of factors, as does 119 (pretty low compared to 120, which is my point -- not that all such positions are prime). Furthermore, looking at the prim 240, both 239 and 241 are prime, and so on.

Incidentally, if you're looking for primes, there's no reason to consider repeated prime factors in these "highly divisible numbers" -- if a number has just a single factor of, say, 2, then the numbers next to it cannot be divisible by 2.

Right, the last thing I quoted you on above is very nearly exactly one of my main points. When a highly divisible number N has the factors p, q, r, s, t, all of which are greater than one, then the numbers N+1 and N-1 cannot have as factors p, q, r, s, or t. This is what I called the displacement principle, and it is why the "prim" numbers, which are highly divisible (having many factors) have as their immediate neighbours (+/-1) numbers which are relatively deprived of factors, and are often completely deprived of them (prime). I didn't say they always are (e.g. 121, which you cite.)

My other point is that the progression of primes and their distribution, far from having their own inherent probability of occurrence, are specified and explained by the probability of n having many factors, which gradually decreases with n (though the number of possible individual factors increases with n-- thus the logarithmic probability distribution). Moreover, another interesting point is that the multiples of 2, 3, 5, 7, etc may be regarded as cyclic or wave phenomena moving up with n, (because they periodically cause a number to have themselves as factors), and when a significant number of them are in phase, you get both I) a highly divisible prim number and II) sometimes, if enough "factor cycles" were in phase on that prim, you get primes on one or both of n+/-1!

Thus it is the prim numbers, specifically their locations, which determine the locations of primes, and explain the phenomenon of primes fully (even the way in which it is the probability of n having multiple factors that decreases by factors related to e, explaining the pi function (the distribution of primes) and its relationship to these probabilities of having factors, 1/ln n.

Isn't that neat? I'm sure it is a new way of conceptualizing the distribution and cause of primes.

 

[marteinson]

By the way, I think this thread has been moved to the wrong place (I meant for it to be in number theory, along with the other discussions of primes).

 

[Hurkyl]

But 37 is next to 36, a highly divisible multiple of six. Also, 121 only has a single pair of factors (pretty low compared to 120, which is my point -- not that all such positions are prime).

The thing you're still ignoring is that the numbers you're considering are TINY. There are very few ways to multiply three things and get something that isn't much larger than 120. 7 is the smallest prime bigger than 5, and 7 * 7 * 7 = 343, much larger than 120.

But when you move to larger (but still small!) numbers, that's no longer a problem.

For example, look at 12! = 479001600. That number is large enough that 782^3 is still smaller than it, so numbers near it could certainly have factors all larger than 11, yet still have three factors.

And, almost prophetically, 12! + 1 = 13^2 * 2834329. (12! - 1 is prime)

13!? Well, 13! + 1 and 13! - 1 both have only two prime factors.

15!?

15! + 1 = 59 * 479 * 46271341
15! - 1 = 17 * 31^2 * 53 * 1510259

15! - 1 has five prime factors, four of them distinct!
16! + 1 has five distinct prime factors. 16! - 1 is not prime.
17! + 1 and 17! - 1 have three distinct prime factors.
18! - 1 has 6 distinct prime factors! (and, again, 18! + 1 is not prime)
19! + 1 and 19! - 1 aren't prime.
20! + 1 and 20! - 1 aren't prime.

21! is just starting to cross over into the realm of numbers that at have a decent size. (But are still by no means large). (My criterion for stopping? It no longer fits into a 64-bit integer, so I'd have to use a large number package, like java.math.BigInteger)


Clearly not all such positions are prime -- as we see, few are prime. As numbers get larger, the spurious patterns you see with tiny numbers disappear.

Right, the last thing you said above is exactly one of my main points. When a highly divisible number N has the factors p, q, r, s, t, all of which are greater than one, then the numbers N+1 and N-1 cannot have as factors p, q, r, s, or t. This is what I called the displacement principle, and it is why the prim numbers, which are highly divisible (having many factors) have as their immediate neighbours (+/-1) numbers which are relatively deprived of factors, and are often completely deprived of them (prime). I didn't say they always are (e.g. 121, which you cite.)

But you missed both of my points. I hope I've made the first clear above. When numbers get large, your observation becomes more and more irrelevant. Who cares that 20! + 1 and 20! - 1 can't have 2, 3, 5, 7, 11, 13, 17, or 19 as a factor? It has all the primes between 20 and 1559776268 as candidates for factors!

The second is that if you're trying to eliminate prime factors, then repeated factors are irrelevant. N having 2^64 as a factor is just as good as N having 2 as a factor when it comes to forcing N+1 and N-1 to not be divisible by 2. I was saying you should look at 2, 2*3, 2*3*5, 2*3*5*7, et cetera... but it suffers from the same largeness problem I pointed out above -- the pattern is spurious and disappears when you get to larger numbers.

For example, if N = 2 * 3 * 5 * ... * 19, then N+1 and N-1 are still both composite. (One has 2 factors, the other 3)

 

[marteinson]

The thing you're still ignoring is that the numbers you're considering are TINY.

Sorry, but I think you're ignoring my point. I generalized it algebraically for you (N having the factors p,q,r,s, and t means that N+/-1 can't have them as factors) -- which cannot be said to apply only to "TINY" numbers.

Also, you misunderstand what I am saying about the factorials (and strangely, you state that they are not the most highly divisible numbers, which they are, by definition -- "factorials" are composed of sequences of factors).

I didn't say that the numbers next to factorials are prime, and I can't figure out what you think I said about factorials. I observed that the numbers one above and one below highly divisible numbers, such as the factorials and multiples of those factorials, are the only place you can find primes other than 2 or 3.

How can you say factorials aren't the numbers with the most factors? 7! = 5040 for instance, which is the 840th multiple of 6, has scores of factors, as your formula will show, but its neighbour 5039 has zero factors (a prime), and its other neighbour 5041 has only one pair of factors that I can see, 71x71. Therefore as we begin to consider larger and larger n, my displacement principle is more and more glaringly visible: In the neighbourhood of 5041, most numbers have long lists of factor pairs, but at 5039, 5041, and 5042, you suddenly get a minimum zero, a local maximum, and another minimum 1, on the relation describing the number of factor pairs n has.

So your assertion that what I am saying applies only to small numbers is wrong, because my generalized "displacement principle" demonstrates that numbers with many factors are adjacent to numbers with very few factors. It seems you don't wish to analyse what I am saying fully. What I am saying is completely true. And it explains why all primes, whatever their size, have zero factors precisely because they have a neighbour which is divisible by all non-trivial (non-1) factors available at that order of magnitude and range of n. I don't understand why you don't see that, and why you answer without thinking through what it means.

 

[Hurkyl]

I generalized it algebraically for you (N having the factors p,q,r,s, and t means that N+/-1 can't have them as factors)

Did you read my post at all?

I gave a theoretical reason why your analysis is flawed: N, and the number of possible primes that could divide numbers near N, grows MUCH faster than the number of factors appearing in your list.

And, I gave an empirical demonstration that your analysis is flawed.


Yes, if p, q, r, s, and t all divide N, then they cannot divide N+1 or N-1. However, your conclusions do not follow from this fact.

I said the numbers one above and one below highly divisible numbers, such as the factorials and multiples of those factorials, are the only place you can find primes other than 2 or 3.

A rather vacuous statement, since prime numbers cannot be found next to an odd number, and all even numbers are multiples of 2!.

How does the fact that 2 * 41 + 1 is a prime number fit into your conjecture? 82 is, in fact, a multiple of 2!, but not any other factorial.

How can you say factorials aren't the numbers with the most factors?

I gave an example.

12! = 479001600 has 792 factors, yet 279417600 has 810 factors! (Both totals are including 1 and the number itself)

279417600 is a smaller number than 12!, yet it has more factors.

And it explains why all primes, whatever their size, have zero factors precisely because they have a neighbour which is divisible by all non-trivial (non-1) factors available at that order of magnitude and range of n.

That isn't even true for tiny numbers.

For example, it's not true of 37. The only prime numbers appearing in its neighbors are 2, 3, and 19. It's missing, for example, 5.

Or 53. It's neighbors only have 2, 3, and 13 for prime factors... but not 5 or 7.

 

[marteinson]

What strikes me, however, is that primes only occur one above or one below (frequently both above and below) the most divisible natural numbers. These are 1x2x3, or 6, and 1x2x3x4, or 24, and so on to 120, 720, etc. Notice that these factorials are the source of e, which describes the frequency of primes in 1/log(n).

It seems emotion and other aspects of human nature are clouding the dialogue, especially since you accuse me of not reading your posts when you haven't read mine very carefully. Where, above, did I include factorials such as 2!, which you use to "prove" my statements are "irrelevant?"

A. Primes ONLY occur one below and one above the most divisible natural numbers (those having the largest proportion of factors). B. They occur in such locations because these most divisible natural numbers, which I called "prim" and are always multiples of six, (though they can also have other further factors) are themselves divisible by all the available factors at that order of magnitude, depriving their neighbours of those factors.

If you were saying my point wasn't new, that would be another story. But you're throwing irrelevant discussions of factorials back at me to say my points are not true.

Aren't you supposed to be a moderator? Was it you who moved my thread to this incorrect location?

Please calm down before contributing to the thread, thanks very much, friend.

 

[marteinson]

I gave a theoretical reason why your analysis is flawed: N, and the number of possible primes that could divide numbers near N, grows MUCH faster than the number of factors appearing in your list.

This is false, unless you misunderstood what my "list" is meant to be: N grows at a linear rate, while the number of possible primes that could divide into it grows at a logarithmic (and therefore slower rate) proportional to 1/ln(n).

For instance, as n progresses beyond 9, the prime factor 3 becomes a possible non-trivial factor that must be discounted in order for n to be prime. As n progresses beyond 25, 6 becomes such a possible factor that must be eliminated, and so on, for 7 beyond 49, etc. Therefore, as n grows steadily, the number of possible factors introduced grows more and more slowly.

Thus it is highly significant that when n is divisible by 2, 3, 5, ... p, q, r... its immediate neighbours, n+/-1, are indivisible by those factors, and can even be prime. Once again, simply put, it is the high divisibility of n that causes its neighbours to be prime or at least have far fewer factors.

Were you talking about some other list?

Also, I still don't understand what you're saying about factorials not being the most highly divisible numbers.

Thanks for your input in any case.

 

[marteinson]

Now I believe I see how Hurkyl misinterpreted what I was saying.

It seems he thought was saying that the factorials, 6, 24, 120, 720 are the only numbers so highly divisible that they can be what I consider 'prim' and adjacent to primes. That is not what I said. All the multiples of these factorials (such as 6, 12, 18...) are multiples of six, and the points I feel are worthy of mention are i) that these numbers, the multiples of six, are highly divisible, ii) this is how I express the fact that they have a large number of distinct factorizations, iii) such highly divisible numbers are in positions which cause them to be divisible by a large proportion of non-trivial factors available at their order of magnitude (between 2 and sq.rt.(n)), always including 2 and 3, and often including other prime divisors, iv) when n is a multiple of six and is also divisible by many other prime factors, such as 5040, you find its immediate neighbours cannot be divisible by 2, 3 or any such factors n is divisible by, so you get n+/-1 both having far fewer factors than other natural numbers at that order of magnitude. They often have zero factors and are thus prime, but even when they are not prime, they show a distinct drop in the number of factors relative to their nearby ranges.

Thus, it is interesting that the probability of n having MANY factors that determines the probability of finding a distribution of primes at n+/-1. This is not contradicted by the fact that n+/-1 often does have factors. All kings are men, but not all men are kings. All primes are one below or one above highly divisible numbers. I didn't say anything about such positions also having composites (though my displacement theory also explains why, when n+/-1 are composite, they have very few factors indeed.

Hope that clarifies what I have been saying somewhat. It is a shame that we all get distracted and diverge from each other when talking about such beautiful and elegant mathematical phenomena.

Cheers.

 

[Hurkyl]

A. Primes ONLY occur one below and one above the most divisible natural numbers (those having the largest proportion of factors).

False. For example, consider the prime 127. It's neighbors, 126 and 128, have 12 and 8 factors, respectively. However, 120 has 16 factors. The prime 127 is not one above or one below one of these "most divisible natural numbers".

How about the prime 257? It's neighbors have 9 and 8 factors, so they're clearly not "most divisible natural numbers". As an example, the nearby 240 has 20 factors. 180, a much smaller number, has 18 factors!


Let's look larger.
101010113 is prime.
101010112 = 2^6 * 7 * 23 * 9803 has 56 factors.
101010114 = 2 * 3^2 * 547 * 10259 has 24 factors.

Clearly not the most divisible. For example, 5040 is several orders of magnitude smaller, yet it has 60 factors!


I just found a good, larger example:

6402373705728106 has a mere 16 factors.
6402373705728107 is prime. (2 factors)
6402373705728108 has 24 factors.

Also,

6402373705728136 has 32 factors
6402373705728137 is prime.
6402373705728138 has 8 factors


The two numbers

Here is an example of a prime number between two poorly divisible numbers! For comparison, 6402373705728000 has 14688 factors! The typical number in this range has around 16 to 72 factors.

Also, I still don't understand what you're saying about factorials not being the most highly divisible numbers.

Maybe that's because you don't define "most highly divisible", so I have to provide my own definition, which I have chosen to be:

A number is to be considered "most highly divisible" if it has more factors than any smaller number.

And I gave an example: 279417600 is smaller than 12!, yet it has more factors. Therefore, 12! cannot be a "most highly divisible" number.

(Though, it is certainly has more factors than the typical number in its range)

This is false: N grows at a linear rate, while the number of possible primes that could divide into it grows at a logarithmic (and therefore slower rate) proportional to 1/ln(n).

n / ln(n) is a lot faster than logarithmic growth. (Which is ln(n))


But yes, as n grows, the number of primes that could be the smallest factor of n grows slowly. However, the number of small factors that can divide n grows even more slowly.

For example:

When N = 100, we have 4 primes that could be the smallest prime of a number near N. But, a number near N can have, at most, 3 distinct prime factors.

When N = 1000, there are 11 primes that can serve as the smallest prime factor of a number near N. However, a number near N can have, at most, 4 distinct prime factors.

When N = 10^8, there are roughly √N / ln(√N) = 1085 primes that can serve as the smallest prime factor of a number near N. However, a number near N can have, at most, 8 distinct prime factors.


The number of primes capable of serving as the smallest prime factor of a number around N grows MUCH faster than the number of small prime factors you can rule out.

For example, when looking at numbers on the order of 10^8, we can choose:

N = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 ~ 10^7

Which let's us rule out 8 of the possible small prime factors for N+1 and N-1. Yet, there are still around 384 other prime factors possible (less than √N), and it turns out that both N+1 and N-1 are composite. (two and three distinct prime factors respectively)


(You've posted twice since I started composing this message, so I'm just going to stop here now that I've finished this thought, and process what you've written)

 

[marteinson]

For example, consider the prime 127. It's neighbors, 126 and 128, have 12 and 8 factors, respectively. However, 120 has 16 factors. The prime 127 is not one above or one below one of these "most divisible natural numbers".

Natural language isn't perfect, so when you're interpreting someone else, it's best to turn it over from every possible reasonable meaning they could be trying to say, because it's not valuable to take a less plausible interpretation and try and disprove that one.

I wasn't trying to say that 126 has more factors than 120. What I was saying, if you read very carefully, was that both 120 and 126 are among the most highly divisible numbers (both are multiples of six, where I indicated these usually fell), and that that is why their neighbours, 119, 121, 125, and 127 have FEWER factors than others natural numbers at their order of magnitude.

Perhaps you should read the article and look at its Excel sheet appendix before being sure what you think I'm saying, because what I have observed is interesting (though you said you saw no evidence of it): Prim numbers can be defined, and their positions determine those of primes. This is not less compelling because I only quickly defined prims here. All the multiples of six are prim, which is a relative quality. Some other numbers, such as 16, are prim, because they have relatively many ways to be divided.

To me it is fascinating that primes only occur next to such highly divisible numbers, which are at the in-phase points of the cycles of prime factors.

 

[Hurkyl]

Just for the record, I've been implicitly assuming you have something to say other than "Any prime that isn't 2 or 3 must be of the form 6N+1 or 6N-1".


The main point I just want to get across, because it's so easy to forget (even by experienced mathematicians), is that when doing number theory, "small" numbers are special, and that "small" according to number theroy is "gigantic" by normal standards.

 

[marteinson]

Yes, I'm saying that such primes above 3 are prime BECAUSE the numbers at 6N have so many factors (though it is also true that often 6N-2 and 6N+2 have the others that 6N lacks). My point, basically, is that the most divisible numbers (those having the largest proportion of the series 2, 3, 5, 7, 11, 13... as factors) can be seen as points of great factor wealth which cause adjacent points to be factor-poor. A way of looking at primes in terms of the highly divisible numbers to which they are adjacent.

Throughout the literature even the greatest mathematicians all say something to the effect that no explanation can be found for the distribution of primes. I am saying that the distribution of clustered factors, and the distribution of "prims" with great numbers of such clustered factors, in fact provides the necessary explanation. The primes are adjacent to them, by virtue of the finite number of available factors being consumed and hoarded by them. The factors-of-n, against n, may be viewed as an interferogram in which maxima, when n contains many many cyclic factors, are in phase, and cause the minima to which they are adjacent. A way of looking at the phenomeon.

 

[Hurkyl]

The thing is, the property is not nearly as strong as you suggest.

I ran an experiment: I generated random numbers in the range 10^8 through 2 * 10^8.

I defined a number to be "prim" if it was divisible by 2, 3, 5, and 7.

Out of 100000 numbers, 5341 were prime.

Out of 100000 numbers that were not adjacent to a prim number, I found 5161 primes.

Out of 100000 numbers that were one greater than a prim number, I found 23225 primes.


I ran another trial, taking "prim" to mean divisible by 2, 3, 5, 7, 11, 13, and 17.

Out of 100000 numbers, 5411 were prime.

Out of 100000 numbers that were not adjacent to a prim number, I found 5253 primes.

Out of 100000 numbers that were one greater than a prim number, I found 29018 primes.


As you can see, a significant percentage of primes are not occurring next to numbers with lots of small prime factors. (Note that the bulk of the difference, here, is simply because the numbers were allowed to range over things that weren't of the form 6N+1 or 6N-1)


Another thing to note is that you've been deceived by 2 and 3. For example, when considering primness, you look at numbers of the form

30 N + 1 or 30 N - 1

because they're adjacent to things that have 2, 3, and 5 as factors. However, these are just as good:

30 N + 7, 30 N + 11, 30 N + 13
30 N - 7, 30 N - 11, 30 N - 13

None of these numbers can have 2, 3, or 5 as factors either.

(Incidentally, I believe number theorists call these things "wheels")


By the way, there is a technical term for numbers that factor entirely into small primes: they're called smooth.

For example, for some reason I want to consider all primes less than 100 to be my "small" primes. Then, any number whose prime factors are all less than 100, e.g. 8051 (= 97 * 83) is called smooth, as is 99!. However, 1111 (= 11 * 101)would not be smooth, because 101 is one of its prime factors.

 

[ramsey2879]

But 37 is next to 36, a highly divisible multiple of six. Also, 121 only has a single pair of factors, as does 119 (pretty low compared to 120, which is my point -- not that all such positions are prime). Furthermore, looking at the prim 240, both 239 and 241 are prime, and so on.


Your logic that less divisible numbers are next to highly divisible numbers seems to be trivial since you acknowleged that a highly divisible number may be surrounded by composite numbers and there is no basis given for determining which "highly divisible" numbers will be next to a prime.

Listen to Hurkyl. He speaks from experience when he notes that patterns found in small numbers (< 10^9) are not particularly useful when dealing with larger numbers. I am myself only a beginner in number theroy, but in my first response to your thread I tried to express in a helpful way my conclusion that your analysis yields nothing that would be deemed an useful or novel idea to others having more experience in this field. Reading more of this thread merely reinforces this conclusion.

It is an oxy-moron to say that the numbers next to highly divisible numbers are generally less divisible. And for those "prims" that you note are next to primes there are countless "prims" that are not next to a prime since you include all numbers divisible by 6 as a "prim". Not to speak of all the countless counter examples given by Hurkyl

 

[krab]

It is an oxy-moron to say that the numbers next to highly divisible numbers are generally less divisible.

I think you mean "tautology", not "oxymoron".

 

[marteinson]

Well it seems some respondents are answering quickly without reading the article and without thinking through what I have been observing. One person says, above "there are countless prims which are not next to primes", which I never suggested wasn't the case.

Maybe it's easier to think of it this way: a factor of 2 is given to every second natural number, a factor of 3 to every third number, a factor of 5 to every fifth number, and so on. These are cyclic phenomena which can be regarded, like wave phenomena, as going in and out of phase with respect to one another. The number of factors of n against n itself is therefore a relation that can be regarded as an interferogram.

What I am saying is interesting is that these phase variations explain the distribution of primes in a conceptualization that is correct and appropriately descriptive: primes occur adjacent to numbers having this in-phase quality, which I have called prim numbers.

Another interesting thing about this conceptualization is that it explains the way in which, because non-trivial factors need only be regarded as coming into play at the square of each prime (25, 36, 49...), which accounts for the ever increasing probability of factor attribution as n increases and the ever-decreasing capacity for the factor cycles to come into phase... thus the shape of the Riemann zeta-function's shape, whose derivative is an expression of this "interferogram." I don't share the opinion that this is tautological, irrelevant or mistaken.

Hope there are some others who find it interesting and compelling too. I've never seen primes considered in this light in the literature.

 

[marteinson]

Your logic that less divisible numbers are next to highly divisible numbers seems to be trivial since you acknowleged that a highly divisible number may be surrounded by composite numbers and there is no basis given for determining which "highly divisible" numbers will be next to a prime.

There is such a basis: when N (a multiple of 6) and N+2 together have as factors the entire sequence of primes between 2 and the square root of n+2, N+1 is prime; when N and N-2 together share that list of primes as factors, in whatever distribution between the two of them, once again, N-1 is prime. N, however, being a multiple of 6, has 2 and 3 as factors, and usually it is "more prim" than both N+2 and N-2 as it has a head start in "collecting" prime factors.

I'm not saying this is computationally a breakthrough, I think it is a better conceptualization of why primes occur where they do: one greater or one less than prims. Why does sharing this conceptualization provoke such a polemical reaction? Also, the moderator still hasn't explained why he moved the thread into general physics from number theory, where I posted it. Can he move it back where it belongs, or does he wish it to be less viewed?

 

[marteinson]

The thing is, the property is not nearly as strong as you suggest.

I am honoured and grateful that you have examined the issue, Hurkyl, but may I suggest you have defined prims in the model above in so restrictive a manner that the property does not seem strong? For me, while primeness is an absolute quantitative property, primness is a qualitative and relative one, indicating that a number has a relatively larger proportion of prime factors among those available between 2 and its square root, depriving its neighbours of those factors. So, when n is "prim enough", especially when n+2 and n-2 contribute their weight by being very prim as well, then n-1 and n+1 are prime, and that in anthropomorphic terms (the way humans conceptualize natural phenomena), primes may be regarded as being caused by the position and degree of the prims.

 

[marteinson]

I defined a number to be "prim" if it was divisible by 2, 3, 5, and 7. Out of 100000 numbers, 5341 were prime. Out of 100000 numbers that were not adjacent to a prim number, I found 5161 primes.

Are you sure of this? I don't see how you can find even *one* prime, much less 5161, not being adjacent to a multiple of six (that would contradict Eratosthenes 6N+-1 observation). If your "prims" were divisible by both 2 and 3, among the others, wouldn't they have to be divisible by 2x3, and therefore be multiples of six? It seems likely your software interpreted your prim definition as being divisible by 2, or 3, or 5, or 7, not "and."

 

[Hurkyl]

There is such a basis: when N (a multiple of 6) and N+2 together have as factors the entire sequence of primes between 2 and the square root of n+2, N+1 is prime

You should note, though, that if N is too large (say, bigger than 300) this is impossible.

Suppose that N is around 300, and that N and N + 2 have between them all factors all primes from 2 through 17.

However, that means N * (N + 2) must be a multiple of 2*3*5*7*11*13*17 = 510510, which would require N to be around 714.

This discrepancy just gets bigger and bigger as N grows.

primes occur adjacent to numbers having this in-phase quality

False. Some primes do, some primes do not. Let's look at two primes I've posted in this thread, and the prime factorizations of the numbers adjacent to it.

6402373705728107 is a prime.

6402373705728106 = 2 * 479 * 1249 * 5350730443
6402373705728108 = 2^2 * 3^3 * 59281238016001

6402373705728137 is a prime.

6402373705728136 = 2^3 * 17 * 421 * 111820135981
6402373705728138 = 2 * 3 * 1067062284288023

Also, the moderator still hasn't explained why he moved the thread into general physics from number theory, where I posted it.

I did not move it, but I do not disagree with it being moved.

 

[Hurkyl]

Are you sure of this? I don't see how you can find even *one* prime, much less 5161, not being adjacent to a multiple of six

I didn't. All the primes I found were adjacent to a multiple of 6.

But I found 5161 primes out that were not adjacent to a multiple of 2*3*5*7 = 210.

 

[marteinson]

All the "objections" and "disclaimers" you have published here amount to narrowing and altering my statement before refuting it. You defined prims as multiples of 210 only, then used this to suggest my observations are incorrect.

By the way, both your large primes quoted very recently above, are not only adjacent to multiples of six, but these are very highly divisible, numbers, and do nothing to discount my observations.

I do agree however, that my idea about having all the factors in N and N+/-2 needs further refinement.

Still, it is a shame people fall over themselves to attack anything new. First, there was "no evidence" for my observations. Later, it was not "as strong" as I seemed to suggest. Why not do as I suggest before critiquing, i.e., take the best possible interpretations of what I am saying and analyse them, instead of refuting unlikely distortions of it? English is a useful tool and I notice all the posts use it here, so I think more care should be taken in interpreting its expressions of people's ideas. Where did I suggest my concept of prims was limited to multiples of 210, for instance?

I'm not satisfied that any respondent so far has made the effort to thoroughly understand what I mean, though efforts HAVE been made to criticize it generously. If you responded with objections, and I have redirected them, would you kindly reply with objections for what I am actually saying?

My conceptualization, for instance, explains existence of the prime pairs, as well as describing the distribution of primes in terms of the clustering of factors in prims. So far, no one has said anything I find addresses these observations directly.

I wish I could do the kind of analyses Hurkyl has kindly attempted, but I would define prims as numbers having all but one of a series of prime factors, then all but two of a series, then all but three of a series, and in all three cases the range series would be extended appropriately according to the order of magnitude of n being considered. Then we would see, I am sure, that the numbers to which primes are adjacent are always considerably more "prim" than others in the neighbourhood which are not adjacent to primes.

Thanks.

 

[WaterBreath]

I hope you guys don't mind the interruption of a newbie. And one not formally educated in number theory, at that...

I think marteinson's conjectures are interesting, but not necessarily groundbreaking. They seem to me rather to be the "next logical leap" after considering that all non-prime numbers have at least one prime as a product. Upon learning this, it immediately occurred to me to think of the number line as a series of superimposed waves with prime periods. Any integer where one of the waves intersects the number line (a node) is a composite number. Any integer where no wave intersects the number line (an antinode) is a prime number.

I think that the displacement conjecture has merit. But again, it seems to follow very logically from the picture above. Take a number k, divisible by m. Any number n, where k - m < n < k + m will, by definition, not be divisibly by m. I envision this as a "ring" (not in the algebraic sense) around k. And when considering many such "rings", it begins to start to look like ripples surrounding a rock dropped into a pond. There is a further comparison, like the one above, that composite numbers occur at points of destructive interference (or nodes) between the ripples surrounding different numbers, and primes occur at points of constructive interference (antinodes).

It's all very interesting and eloquent or elegant to consider these visualizations. But that does not change the fact that what determines the "knots" of nodes, where an integer happens to be divisible by many primes, is in fact the distribution of those primes. So we have somewhat of a recursive definition here. What determines the distribution of "large" (as Hurkyl put it) primes? Why, the distribution of smaller primes, of course. And what determines the distribution of those? Even smaller primes. And this can be continued on until you're down to "really small" primes such as 2, 3, 5, 7, 11, etc.

But none of this really sheds new light on how to quickly determine the next prime, or whether a particular random number is prime or not. The best we can do, without recursively examining all the primes starting from the beginning, is assign a probability based on a series of more sophisticated calculations.

Anyway, just some thoughts. Hope I haven't wasted anyone's time.

 

[Hurkyl]

but these are very highly divisible, numbers

No, they're not. Most numbers of they're magnitude have 3-5 distinct prime factors.

 

[marteinson]

It's refreshing to see the Waterbreath is looking at my point with an open mind... he is also visualizing the prime cycles as wave phenomena, as I suggested, and has understood that at their nodes, when they are in phase, this is where we find natural numbers with especially high numbers of factors.

I'm not entirely sure I see his opinion that the definition is recursive in the same way he does, however... one could say that most definitions in math are tautological, recursive, or self-evident...

It's a chicken and egg concept. I see the distribution of primes, which is nothing other than the distribution of natural numbers with no factors, as a function of the distribution of *factors* among the natural numbers, and I noticed that the primes, and even other numbers in similar positions, are deprived of factors by the factor-rich "nodes" or in-phase points.

It's so predictable and mechanical using this conceptualization that you could actually build an apparatus out of wooden wheels that could crawl along a plane of natural numbers designating those which are prim and prime.

As for Hurkyl, I still think he ought to re-do his 'test' without being rather obtuse towards my thesis. The fact that primes occur next to, and between, highly divisible numbers is an observation which, once seen, is theoretically evident and can't be attacked by bizarre little examples of ambiguous pseudo-exceptions.

Like I said, when n has the factors p, q, r, s, t... and others, and when this series of factors in n constitute a significant proportion of those available at the order of magnitude of n, we get n being highly divisible and relatively "prim," and we get n+1 and n-1 showing themselves as particularly factor poor, often prime.

A few interesting discreet examples, however impressively large, can't refute this, which is useful because it provides i) a useful conceptualization of primes and their distribution, ii) an explanation of primes which is sufficiently descriptive iii) a reason for the existence of twin primes and iv) a theoretical confirming of the observed Li(x) function for the decreasing frequency of primes as x increases, and it also v) provides a conceptualization explaining why the reciprocal of the natural logarithm function is the correct smooth function giving the probability of finding primes at n, since its derivative decreases at the rate new non-trivial factors are introduced with increasing n.

I thought I would share it with you, and didn't claim it was groundbreaking, though I think it's a step forward in conceptualizing and explaining in the natural science sense of "erklaren" or shedding light on something.

In essence, I'm saying it's not the primes which are "distributed" where they are, but the factors which are distributed, and the patterns according to which the factors are distributed have distinct properties that help conceptualize and explain what primes really are and why their positions are deprived of factors.

 

[shmoe]

I think that the displacement conjecture has merit. But again, it seems to follow very logically from the picture above. Take a number k, divisible by m. Any number n, where k - m < n < k + m will, by definition, not be divisibly by m. I envision this as a "ring" (not in the algebraic sense) around k.

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).



By the way, ideas relating to the conditional probaility of a number being prime given that it's neighbours have *whatever* as factors have been used for years. See the twin primes (and related) conjectures by Hardy and Littlewood, work on prime constellations, prime number theorem for arithmetic progressions, etc. The last is particularily relevant, if you randomly pick a "large number" that's relatively prime to m, your probability of getting a prime will be about m/phi(m) larger than if you just picked a random number (using the same version of "large"), where phi is the usual euler-phi function. If m is the product of primes less than x, this ratio is asymptotic to $$e^{\gamma}\log{x}$$. It says actually something much deeper than this and you can go into specifics about the probability a number am+r, where 0<=r<m and a is "large", is prime based entirely on r.

That's really all I have to say. This is plagued by the same problems the last time marteinson posted it, ill defined terms, vague language, an extremely small set of data, and the still very humerous "I leave the formally correct proof to real mathematicians, however" [refering to the Goldbach conjecture].

If you are really interested in doing any mathematics, you should learn how to make precise definitions and make unambiguous statements. It's frustrating as can be to have to try to come up with our own seemingly resonable definitions for your vague notions, and then be told that's not what you've meant at all and that people just aren't reading what you've written. I have no idea what "..but I would define prims as numbers having all but one of a series of prime factors, then all but two of a series, then all but three of a series, and in all three cases the range series would be extended appropriately according to the order of magnitude of n being considered.." is trying to say. I can't guess as to exactly why a moderator has decided to move this to TD, but this sort of thing is enough to get my vote (not that my opinion on this matters).

 

[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.