Partner to the Prime Numbers: Highly Composite numbers, or, Quantitative Divisorial Initiation

Revised Sept 2013

On The Takeway (12/12/12), John Hockenberry noted that 12 has 'lots of factors or divisors. Well, it has the most factors of any smaller number, as has only multiples of 12 than has any number smaller than that. I call this 'quantitative divisorial initiation' or QDI for short. 12-and-its-QDI-multiples is the ONLY family of integers that has QDI (for any number which is NOT a multiple of twelve, it never has QDI: none of the numbers which are not multiple of twelve are QDI numbers, since some smaller number which IS a multiple of 12 initiated the quantity of divisors which that larger number merely replicates.

 

In other words, a number which has a greater quantity of divisors than any one of its predecessors is a number which thereby can be described as having 'initiated' the increase in quantity of divisors. No two QDI numbers are contiguous, but are rather separated by at least twelve other numbers. So, while in many cases the yard, or separating space, contains numbers which equal the quantity of divisors of the lesser QDI, no gap number exceeds the lesser QDI in terms of quantity of divisors. And, since the minimum gap is twelve numbers long, and since every QDI is a multiple of twelve, there is a lot of variation in quantity of divisors between any two 'next door' QDI numbers but never more than that of the lesser QDI.

3 and 6 are the only single integers with QFI. Algebraically speaking, 6 has two factors (i.e., 'distinguishing factors', since 1 and n are common to all numbers), and no number smaller than 12 has more that two distinguishing factors.
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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 with the most exquisite skill carrying out the most demanding of 'marching orders'.

For example, by parsing the natural numbers in series of twelve beginning with 1, with 13 being counted as 1 in the next series of twelve, all primes above 12 occur only in the series-nominal positions of 2, 6, 8 or 12.

But, other than these kinds of these finely tuned 'marching orders', the prime numbers also are extremely unpredictable: they occur or not occur without strict one-to-one predictability. Primes are like the 'Rambo of numbers.'

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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.

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 QDI number over 1, and also as reflected by 6 as the highest of the smallest non-primes with a continuous string of factors:

 

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 (not quite ad good as 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!!!). 

35 is divisible by 1, 5, and 7 (AAAHH! Another 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 or 35 each, and a jump of +2 more than for 34 and 35 put together!).

 

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. The 'grid' to which I refer is keyed to those four continuous factors; So, for example, the sum of the factors of 12 is 10, and the sum of 10-and-12 is 22.

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. Even 3 can be grouped as a set of (2 and 1), but 2 can be grouped only as a set of (1 and 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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