B U T T E R F L Y   P R I M E S
~P r e j u d i c i a l   N u m b e r s~
~let the beauty seep in~
~metamorphosis~
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Butterflies take awhile!
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N U M B E R S
the ultimate prejudice

~take all the ordinary numbers and get rid of the redundant ones.

Redundant ones? What redundant ones? You mean some numbers aren't as pure and worthy as others?
~yes, and though all numbers have some quirks...some tricks up their sleeves..some personality so to speak...let's face it the even ones are just redundant. they're just made by doubling their odd neighbors or by adding up a couple of prime numbers. (Prime Numbers, you know, numbers which are only reducible by themselves and one, and which are all odd, too, except two.)

So evens are out and odds are in?
~yes, but we are going to keep two sets, two and twenty-four related evens. right now, let's get rid of most of the odd numbers.

What! What numbers are going to be left? How do we know which odds go? Don't all these numbers, even the even ones, have a reason...a purpose...for their existence? Isn't their presence part of the whole, the beauty of creation, the expansion of the harmony, richness and diversity of the universe? Couldn't we just as easily say that all the natural whole numbers are even and the odds are simply the result of division by two? Just who decides where this prejudicial knife is going to cut?
~trust me. Nature in all her glory and profound richness nevertheless builds diversity upon the scaffolding of simplicity...efficiency...and the path of least resistance. throw in the ability to add options...an even number here, an odd there...allow for seemingly random, chaotic influence (the flap of a butterfly's wing) and you have a recipe for richness and diversity. that said, it's simpler, more efficient and the least resistant path to start with 1,2,3,5,7,9,...rather than 2,4,6,8,10...

~because we are being prejudicial here. we are looking for the one! back to the odds: get rid of all of the redundant odds that can be formed by adding or multiplying together two or more primes... leaving behind all the primes, which are all odd, except for two.

Seems to me you have trapped yourself. What about one?
~your knife is sharpening! yes, one times one times one, etc. is still one. so one must not be prime! in fact, our final act of prejudice is to eliminate all "prime" numbers which can only be reduced by themselves and one...two different numbers...leaving only one. and because one is such an elemental number which can form all natural whole numbers by simply adding itself, we will call it not "prime", but "unit." one is a unit and it is the one! you are the man! you are the one! one is you! go girl! you are woman! you are the one in a million gazillion! leave one alone. "one is the loneliest number..."

Cute!? So that's it, your big prejudicial analysis leaves us with the boring unit one...a seemingly random list of primes...an even longer list of odds and evens (at least they have order). Isn't there more?
~you're right. it seems our barren landscape of pure primes numbers is, by itself, almost meaningless unless you are prime to something. actually, for over two thousand years we have been adding to a very long list of mathematical wonders involving primes. of course, in hindsight, it is only natural that if all the numbers (greater than one) are made of primes, that primes are going to figure into countless mathematical niceties...you know, expressions of growth, creativity, love, blah, blah, blah.

Ok, so we know that...or at least we can look that up...what about the pattern of primes themselves ...hasn't anyone found an order...a predictable pattern...to support Riemann's Hypothesis (that there is an order)?
~funny that you asked. it's called "Butterfly Primes". undoubtedly, it's just a beginning. it turns out that the set of twenty-four-related even numbers is the key. on a multiplication matrix (table) with one at the origin, highlighting all the n24s within...that is all numbers which are multiples of 24...a reproducible pattern of highlights emerges on all four quadrants of every n24 square. removing the upper right and lower left quadrants in each case leaves a pattern of n24 highlights, with a diagonal line of "squares"...including the squares of all the prime numbers.
~not only is the difference between the squares of any prime (above 5) always a multiple of 24, in every occurrence of the squared prime on that diagonal, but we also see that it is always shepherded by two n24 highlights that are complements to each other [e.i. 48=6x8=8x6=n24=(2)(24)]. "Butterfly Primes" have emerged.
~said another way: take the square of any prime (greater than 5) and subtract one. the resulting number is a n24 number that is always formed from multiplying two even numbers which are exactly two apart [e.i. 172=289 and 289-1=288=16x18=18x16=(12)(24)].

Well take off the blinders! Anyone can see that you're little n24 Butterfly pattern incorporates all the even numbers!
~let's take a look. hmmmm. yes, you are quite right. the full Butterfly pattern (outline and internal n24 highlights) requires all the even numbers...although a lesser number qualify as the shepherds of the primes.

So if the evens are back in the game, what about the odd odds...you know, the odd numbers (greater than one) that are not prime, like: 9,15,21,25,27,33,35,39,45,49,51,55,57,63,65,69,75,77,81,85,87, 91,95,99,105,111,115,117,119,121,123,125,129,133,135,141,143,145,147,...?
~i thought i made it clear, odd numbers that are not prime are just redundant sums and products of primes. they deserve little more than that, don't you think?

I think that you are off your rocker!...but that's neither here nor there. What if you were to take all the odd numbers, square them, and subtract them individually from all the other squared odd numbers, and then divide the individual answers by 24 (just like with the primes above). Then plot out on a matrix grid those "n" results...that is, just the "n" value. What do you find?
~hmmmm. well, it seems a new and strikingly different distribution pattern emerged.

How so?
~plotting out all the "n" values on a matrix grid, with the odd numbers on the x- and y-axis, and 1 at the origin, shows regular alternating bands of two sets of numbers. one band, which occurs regularly at every third odd number (9,15,21,... starting with 9), is composed exclusively of non-prime numbers...all of which are evenly divisible by 3, their row sums are divisible by 3, and they always have a positive "n" value in the first horizontal position of each row under the 32 position. Call it the Blue Group.
~the other band contains sets of numbers laid out in two rows...the numbers of which always line up in straight columns of two, with a blank space between, giving a distinct 4x4 square pattern of numbers. These alternate between the Blue Group or Bar rows.

~well, some rows (and their corresponding columns) are made from the prime numbers and some are not.

I think that if you look a little closer you will find that all the numbers in this other second set between the Blue non-primes are either prime or "prime-like", none of which are in the Blue Group.
~what do you mean when you say "prime-like?"

"Prime-like" numbers are either the products of primes or they are numbers that, while not prime, are divisible by 5 (and do not overlap the Blue Group). They both mimic the behavior and placement of the actual prime numbers and their patterns within this second set of numbers.
~so prime reigns supreme again?

Try and look at it this way and see if perhaps your number prejudice is not in fact misguided... or at the very least...inappropriate in the grander scheme of things. You know, as in the whole is greater than the sum of the parts.
As Nature, you build from the simplest numbers and their combinations to provide redundancy, stability and expansibility. You start with the unit 1. Double that to get 2. Combine that with 1 to get 3 and you have structure (and the magic of 5 results from 3 +2). You lay down a scaffolding framework of every third odd number (starting with 9=3x3) which of course is never prime. In between these ribs, you lay in those numbers which are either prime or "prime-like". First the "prime-likes", as those non-Blue Group non-primes which are either divisible by 5 or are multiples (products) of primes. What remains are the primes. The primes are the numbers leftover when all the redundancy and regularity and expansibility of all the odd (and even, by the way) numbers are filtered out.

~so you are telling me that the primes are special by default and not by design?

No!...you're still not getting it...it's all by design. All the numbers are special. The prejudice is not in our perception, but in our conception.
~what do you mean?

All the numbers are special. The prejudice is not in our perception that there are differences in the numbers and the various patterns and permutations that they can make, but in our conception that this difference for this number group is more valid...deserves more favor...than that of another, where in fact, the very existence of the one is built and predicated on the existence of the other. It's all by design. The negative space of one is defined by the positive space of the other!