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Of Primes, prims, pairs and e

  1. Jun 6, 2005 #1
    "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

    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

    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
    Last edited: Jun 6, 2005
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  3. Jun 6, 2005 #2
    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 .
  4. Jun 6, 2005 #3


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    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)
    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.
    Last edited: Jun 7, 2005
  5. Jun 7, 2005 #4
    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,


    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:




    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! :cool:

    This should be a reply to original poster..!
    Last edited by a moderator: Apr 21, 2017
  6. Jun 7, 2005 #5
    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.

    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.
    Last edited: Jun 7, 2005
  7. Jun 7, 2005 #6
    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).
  8. Jun 7, 2005 #7


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    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! + 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.

    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)
  9. Jun 7, 2005 #8
    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.
    Last edited: Jun 7, 2005
  10. Jun 7, 2005 #9


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    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.

    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.

    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.

    :confused: 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.
    Last edited: Jun 7, 2005
  11. Jun 7, 2005 #10
    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.
  12. Jun 7, 2005 #11
    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.
    Last edited: Jun 7, 2005
  13. Jun 7, 2005 #12
    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.

  14. Jun 7, 2005 #13


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    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.


    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.

    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)

    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 lets 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)
  15. Jun 7, 2005 #14
    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.
    Last edited: Jun 7, 2005
  16. Jun 7, 2005 #15


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    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.
    Last edited: Jun 7, 2005
  17. Jun 7, 2005 #16
    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.
    Last edited: Jun 7, 2005
  18. Jun 7, 2005 #17


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    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 occuring 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.
  19. Jun 8, 2005 #18
  20. Jun 8, 2005 #19


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  21. Jun 8, 2005 #20
    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.
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