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→Beyond 930
== Beyond 930 ==
== Beyond 930 ==
...to be posted later...
Prime numbers are those with the least possible quantity of factors/divisors (respectively themselves, and 1). In that sense, and in many other senses, they are very much predictable, like soldiers of the most exquisite skill carrying out the most demanding of 'marching orders'. But, despite these finely tuned 'marching orders', the prime numbers also are extremely unpredictable: you never know exactly when or where they will turn up.
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FYI: I believe there is relative utility within any set of algorithms intended to more-or-less predict, or find more-or-less predictive patterns for, the occurrence of the prime numbers: not all such algorithms are equally useful to that intent. In other words, I believe that, given all the various attempts, to date, to predict the primes, there is a general narrowing down of the predictive power of algorithms so intended. But, I believe there logically can never be any sub-set of all algorithms-so-intended that can actually strictly predict the primes. Which means I believe there is no finite maximum of such algorithms that can perfectly predict the primes.
...But, there is a reverse side of that coin...
...namely 12.
Let me explain.
Twelve, or any multiple of twelve, is always the first in the natural series to have an increase in quantity of factors over any lesser number. This implies that any greater number lesser than the next multiple of twelve never has an excess of that quantity of factors. I call this Quantitative Factoral Initiation (QFI).
So, twelve and its multiples are numbers that 'mother the most children', for which the factors of those numbers are the children. This is reflected by 2 as the root QFI number over 1, and also as reflected by 6 as the highest of the smallest non-primes with a continuous string of factors:
QFI is at every sixth number-divisible-by-two, and that sixth number is always a multiple of 12 (rather than of 6):
2 is divisible by 2 and 1.
4 is divisible by 2 and 1 (into 2+2 and 1+1+1=1).
6 is divisible by 2, 1, and 3 (This is the first progress: Per 3a., above; and, Per that one of these three options is a dynamic prime, specifically, the First Dynamic Prime, per 3b. above).
8 is divisible by 2, 1, and 4 (same quantity of divisors as 6).
10 is divisible by 2, 1, and 5 (same quantity of divisors as 6).
12 is divisible by 2, 1, 3, 4, and 6 (Aha! The first ‘large’ number).
14 is divisible by 2, 1, and 7 (Uh-oh! NOW we’re SINKING!).
16 is divisible by 2, 1, 4, and 8 (no better than for 12).
18 is divisible by 2, 1, 3, 6, and 9 (no better than for 12).
20 is divisible by 2, 1, 4, 5, and 10 (no better than 12, so the sharks still circle).
22 is divisible by 2, 1, and 11 (My Gosh! Have we forgotten even how to tread water?!).
24 is divisible by 2, 1, 3, 4, 6, 8…and 12 (Yeeeaahh! A leap of two-dolphins-more-than 20’s zero dolphins, to give us a ride to shore!).
26 is divisible by 2, 1, and 13 (The dolphins have dumped us amidst a veritable horde of sharks!).
28 is divisible by 2, 1, 4, 7, and 14 (no better than for 20, sharks still circle).
30 is divisible by 2, 1, 3, 5, 6, 10, and 15 (no better than for 24, but at least now we’re back on the dolphins, heading for shore).
32 is divisible by 2, 1, 4, 8 and 16 (no better than for 20, the dolphins have again dumped us off).
34 is divisible by 2, 1, and 17 (no better than for 6! AHH! A shark just brushed our leg!!!).
36 is divisible by 2, 1, 3, 4, 6, 9, 12, and 18 (that’s a jump of +5 more than for 34!).
This reflects the root partnership between 2 and 6. So, beginning with 12, every sixth number-divisible-by-two is a QFI number.
QFI permits an exact 'grid' for the entire natural series, in terms of QFI=6 as multiples of twelve. Compare this to a 'grid' using any other multiple, say, multiples of 5 or 10: such an alternate 'grid' is a more arbitrary, less-interrelated, 'grid', partly because, unlike twelve, 10 lacks a continuous string of factors (1, 2, and 5 for 10; whereas 1, 2, 3, and 4 for 12). Any number with a continuous string of at least four factors is divisible by 12, but not any number with a continuous string of three factors is divisible by 10 or 5.
So, 12 and its multiples, per QFI, are the perfect compliment, or counter-balance, to the Principle of Primes (including, but not limited to, minimal factorization), with 2 being the root-in-common between primes and non-primes, and with 6 being the mediator between 2 and 12.
Now, if my ‘grid’ theory is correct for all this, then the only question is: What is the algorithm that defines that ‘grid’? I think it must involve a ‘dance’ between QFI and the primes.
'''A NOTE ABOUT THE ROLE OF 2'''
'''1a.''' Any number which is divisible only by ‘itself and 1’ is called ‘prime’ because a number which is ‘divisible only by 1’ is a tautology. I repeat, it’s a tautology. In other words, to say that A=A is no more informative than simply saying ‘A’.
'''1b.''' This implies, firstly, that any number which is divisible by 1 also is ‘divisible’ by itself (i.e. resulting in ‘1’, that is, in a single instance of the original number---as if that’s not IMMEDIATELY obvious).
'''2a.''' Secondly, it implies that any number divisible by 2 (including 2) is a number a property of which is the ‘Principle of Division’, distinction, dichotomy, basic logic, learning, symmetry, intelligence, etc..
'''2b.''' Any number greater than 2 which is divisible by 2 also is divisible by at least one other number, namely, half the original number.
'''3a.''' But, per 1a. and 1b., above, numbers which are divisible by 2 also are divisible by 1 (which is tantamount to the original number). This means that, for ‘large’ numbers which are divisible by 2, they are divisible by at least two other numbers: half the original number and 1 (with the original number being simply an alternate expression of that 1).
Therefore, I observe that:
'''3b.''' 2 is divisible by 2, and 1, in that it is made of exactly two 1’s. But, 2 is NOT divisible by any other number (i.e., it cannot be otherwise divided into equal units). So, per 1a and 1b, above, 2 is not a ‘dynamic’ prime, but merely is the simple, ‘static’ prime. In other words, 2 has no alterative, unlike all other numbers which are not 1. And, unlike all other primes above 1, 2 is divisible by 2, which serves to further establish the ‘Property of Division’ noted in 2a.
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== References ==
== References ==