Author – Andrew J Frost 10/06/2020 REV TWO PnonP p10
PRIMES AND NON-PRIME PATTERNS -10
OBSERVATION 10 – Complementary Series to Zeta
Putting the primes into the background for a moment and considering the Set of non-prime odd-composite specially defined numbers produced as part of my special Set Saf.
One wonders if it is actually the concept of this Set of numbers, whose complementarity is the key to prime unpredictability and not the prime series itself. Or to put it another way; does the zeta function and Riemann Hypothesis have an opposing operator or “anti-result”, for these numbers; that could be shown and predict the positioning of primes?
Could there be a paper “On the number of non-prime odd-composite complementary numbers less than a given prime”?
That is the gist of the last post PRIMES AND NON-PRIME PATTERNS -09
Interesting Paper
Also, as you can see below (Work By Others) that I found a paper that is very interesting, which I bring to your attention. Remembering here that THIS collection of posts I am writing is a new revised version of previously published information, which is my full paper.
I previously and first published this on my former version of this website on 16/01/2020 and in Facebook on 11/01/2020.
I first started researching patterns to do with Primes around about the beginning of March 2018.
WORK BY OTHERS
Found this paper on the internet 07/08/2018.
Paper by: – Sandor Kristyan (on Arxiv.org)
Sandor Kristyan -August 30, 2017. Hungarian Academy of Sciences, Research Centre of Natural Sciences. “On the statistical distribution of prime numbers, a view from where the distribution of prime numbers is not erratic.”
https://arxiv.org/pdf/1709.02439.pdf
This paper is of course mathematically in advance of what I am illustrating in my posts here on 15711.org.
One of the equations in Sandor Kristyan’s paper is noted as equation 2.
Specifically I’ll call his notation of n = 2(2mt+m+t)+1, “Equation 2”; changed to my notation, n = 2(2ab+a+b)+1 where a>b, or “Eq2” from now on.
Sandor Kristyan’s Equation 2.
Sandor Kristyan’s equation number 2, Eq2, n = 2(2ab+a+b)+1, where a>b; he shows that it generates a Set of prime and non-prime odd-composite numbers.
There is also a lot of other data, including proofs about numbers.
He works with all the prime factors (including 2, 3, and 5), which means he does not consider the specific odd-composite number results caused by a specific integer series (eg. My List) together with the prime results. He certainly does not consider them as one (graphical) entity.
Sandor Kristyan develops the polynomial equation Eq2 and also a number of other equations as well. However it is Eq2 that produces exactly the same resultant list of numbers as my Logical Process List, first described here in PRIMES AND NON-PRIME PATTERNS -01 and -02; but this only happens if the procedure below is followed (‘Patterns within Set Saf‘).
Currently equation 2 produces primes and non-primes, with some of these numbers produced as duplicated or triplicated results; when the whole number line is used as integers for a and b in the equation. Specific results that I require can only be produced by a specific integer sequence.
Sandor Kristyan does not consider 0, zero; as a viable integer for one of the values a or b in Eq2.
Patterns within Set Saf
If the value of a =0, and then a specific integer list is used for b, where b = 3,5,6,8,9,11,14,15….., the series increasing by the repeating sequence of gaps between the integers from 0 +3 = 3, thus 3,2,1,2,1,2,3,1,….., (or the increase of +15 back referring to the first value in the sequence of eight integers shown as ‘b=’ above, to produce the next eight integers thus: – 18,20,21,23,24,26,29,30…….and so on); produces a series of numbers exactly the same as Set Saf.
The Set Saf, consisting of primes and non-primes; specifically non-primes that are the product of previous primes in the list. This is the Set of numbers that I am analysing for patterns within the sequences of this series of numbers.
My List, Set Saf
Set Saf is the Set that contains 2 further Sets of numbers. One is the Set of all primes starting at and including number 7 (Set Sc1); the second is the Set of special non-prime odd-composite numbers (Set Sc2).
Remembering here that 1 is included in the first Set; but 2, 3, and of course 5 are excluded, because they are not considered true primes for this exercise; we are only interested in this Set containing numbers equal to or greater than 7 plus additionally 1.
Contained Set One (Set Sc1) is Completely Complementary to Contained Set Two (Set Sc2).
The “Contained” Sets are: – Sc1 the Set of Primes and Sc2 the Set of Non-Prime Odd Composite numbers.
Contained Sets Sc1 & Sc2
Set Sc2 is exactly complementary to Set Sc1. This relationship is not simply Sc2 + Sc1 = Saf , but that the two contained Sets have a special relationship in that they fit together like a glove, or as I have pointed out in a previous post – like two sides of a zip. This is an infinite relationship as well.
No matter where you look in Set Saf this relationship will be seen to be maintained.
Refer here to a recent post – “Complementarity – Ratio – 3”; the last paragraphs under the heading, “Contained Set Sc1 & Sc2” in the link below which refers to the contained Sets.
https://15711.org/complementarity-ratio-3/patterns/2021/06/02/
Returning to the subject of “Patterns within Set Saf” above
As indicated the sequence of the integer patterns is derived from the pattern of gaps between these integer numbers.
See again the tables showing the relationships of these gap sequences in previous post number 7, “PRIMES AND NON-PRIME PATTERNS -7”.
https://15711.org/primes-and-non-prime-patterns-07-pattern-gaps-between/patterns/2021/03/23/
You can review all other posts in this series on 15711.org. This series is “PRIMES AND NON-PRIME PATTERNS” from number -01 to -10 currently, with more Observations to be added.
Please use Contact Form below if you wish to comment: –
