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Author – Andrew J Frost 05/05/2021 PnonP

Complementarity of Contained Sets

The new spreadsheets are created as follows: –

They are in my Mediafire account – link provided below: –

https://www.mediafire.com/folder/eqq1sbauqhdhu/Spreadsheets+8no.+for+Ratio+P&PnP

Current Data

Currently these eight spreadsheets list approximately two million five hundred thousand rows of data. This information demonstrates a number of observations about the primes and non-prime odd-composite numbers.

• Initially the ratio number is a number to 16 decimal places. This is a natural maximum of decimal places occurring in the series.
• In spreadsheet 1. there is an image of a graph (similar to a rational or logarithmic shape) of the the first few hundreds of thousands of numbers, which demonstrates a curve starting from less than 1 that rapidly increases in value.
• As the numbers increase in size the graph curve slows progressively into a much flatter line that settles to a gently increasing slope appearing almost flat. Graphically it is of course not completely horizontal and maintains its gradual increase in value. The graph has been rotated clockwise by 90 degrees so that it displays vertically at the right hand side of the spreadsheet.
• Also spreadsheet 1. the ratio number passes through 1 at row 1314 (1305 if discounting the first 9 rows that make up the header). The ratio reaches the whole number of 2 at row 58757 (58748 if discounting the first 9 rows that make up the header). At the end of this sheet at row 200040 (200031 if discounting the first 9 rows that make up the header), the final value is of the non-prime 750113, which has 139840 non-primes lower than this number. The last difference available on this sheet is 79647 giving a ratio of 2.32328166276230000.
• It then takes the next seven spreadsheets of the next 2306849 rows collectively (2306840 rows if discounting the first 9 rows in spreadsheet 1, that make up the header) to reach and finally pass through the whole number 3 in spreadsheet 8. Passing through the whole number 3 happens in spreadsheet 8 at row 2506889 (2506880 if -9 first rows). (Actual spreadsheet row number on this sheet is 6931).
• The last prime in spreadsheet 8 is 10874947, and non-prime is 10874951. This means that there are 718059 primes lower than non-prime 10874951 and 2181924 non-primes lower than 10874951. This produces a difference of 1463865 and therefore final ratio of 3.03864167150610000.
• The number of decimal places required now to show the ratio number is reduced by 2, down to 14.

This link takes you to the post called – Complementarity Ratio – THE RATIO OF NON-PRIMES TO PRIMES – https://15711.org/ratio-of-non-primes-to-primes/patterns/2021/05/05/ which explains some of the lead-in detail for the need to construct the eight spreadsheets above.

Patterns

As yet I have not examined the data in these eight spreadsheets for patterns. The obvious patterns are of course maintained (i.e. the: – Ending Number Pattern, Gap Patterns, List Construction Patterns, etc).

My Logical Process Formula

– to recap on the number series being examined: –

• The series is extracted from the number line and is denoted as ℤ. in mathematics, which represents the list of Integer numbers.
• The prime and non-prime odd-composite numbers in this list are whole numbers without any decimal expansion; it is the ratio number as noted above that has decimal expansion to initially 16 decimal places, falling to 14 places.
• This list is My Special List which as a mathematical Set I have called, Set S_af. The list is formed by the process described in: – My Logical Process Formula.

My logical process formula generating my special Set S_(af) comprising prime and non-prime odd-composite numbers: –

12n + {1,5,7,11} = p_(p&np)

and p_{(p&np) –}[div5] = S_(af)

where: –

• n is an integer
• 1,5,7,and 11 are prime numbers used to raise the resultant of 12n to “p_(p&np)” which result in primes and non-primes evolved from odd-composite numbers.
• [div5] is redundant information which in this case is odd-composite numbers ending in 5, hence the ‘minus’ which represents a sieve (or sort) to remove them.

The formula dictates that the integer “n” should be multiplied by twelve and then the product should be added to by 1, 5, 7 or 11. The result of this addition being four numbers that are either a prime or non-prime odd-composite numbers.

The second part of the process is p_{(p&np) –}[div5] = S_(af) which produces My Special List as a Set of numbers that excludes all numbers ending in 0 or 5; this then is My Special Set, S_af, in which immutable patterns are produced.

Set S_af

Set S_af is the Set that contains 2 further Sets of numbers. One is the Set of all primes starting at and including number 7 (Set S_c1); the second is the Set of special non-prime odd-composite numbers (Set S_c2).

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 S_c1) is completely complementary to Contained Set Two (Set S_c2).

The “Contained” Sets are: – S_c1 the Set of Primes and S_c2 the Set of Non-Prime Odd Composite numbers.

Contained Set S_c1 & S_c2

Set S_c2 is exactly complementary to Set S_c1. This relationship is not simply S_c2 + S_c1 = S_af , 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 S_af this relationship will be seen to be maintained.

The special relationship of the second contained Set S_c2 is that it is composed of “elements”, numbers that are the product uniquely of at least two or more primes, listed in the first contained Set _Sc1. (Note – “elements” in this context are numbers but could be a collection of anything).

• Each element of Set S_c2 (odd-composites) contains more than one element of Set S_c1 (primes).
• The product of two or more elements of Set _Sc1 uniquely produce one of the elements of Set S_c2.
• Each element of Set S_c2, is a unique product combination of elements of Set S_c1.
• Elements of Set S_c1 are primes but also they are discretely factors of each element of Set S_c2.
• The elements of Set S_c1 appear in magnitude order that exists naturally within Set S_af and the number line ℕ₀.
• The resultant of the product combinations within Set S_c2 using elements in Set S_c1 (factors >=7), appear in magnitude order that exists naturally within the number line ℕ₀.
• The elements of the two infinite Sets, S_c1 and S_c2 do uniquely intersect at an infinite number of instances. Single or multiple collections of the elements of S_c1 represent a sub Set collection of S_c1, of which the resultant product is equal to a unique element of the S_c2.
• The infinite Set S_c2 is larger than the infinite Set S_c1. The ‘Ratio’ is a direct measurement of size difference of this infinity.

There’ll be more.