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Author – Andrew J Frost – 06/03/2021

The Ratio of Non-Primes (with factors >=7) and Primes (>=7) [but including 1]

I have added a further column to the Main spreadsheet. This column is called “Ratio of Non-Prime to Prime”, it has a background colour of light blue.

Ratio of Non-Primes and Primes, The New Spreadsheet

Due to the spreadsheet size needed, I have had to build a new spreadsheet. It is in https://www.mediafire.com/folder/gysmfq0utwmsb/Spreadsheets+for+Ratio to show this ratio.

This spreadsheet has half a million rows of data.

It’s pretty big (81 Mb).

In This Spreadsheet

In this spreadsheet the column titled “Com.b the Gap columns” is also called Col.1 and shows the number of the gaps combined into one column. These gaps are shown in the two previous columns titled “Prime Gap to Prime jumps NonP” and “NonP Gap NonP”.

These two previous columns show the gap from Prime to Prime jumping over the Non-Primes and the gap between Non-prime to Non-prime jumping over the Primes. This data (in two rows) is adjacent to “the point of examination”.

The point of examination is where the cell formula finds data of the ‘number of primes below the next non-prime’. This point is where the resultant ratio is displayed in Col.5, entitled “Ratio of Non-Prime to Prime”.

Abbreviated names are used for convenience; these names are “Col.1”, “Col.2” and “Col.3”. The Difference is “Col.4” and the Ratio is “Col.5”.

The formula in the spreadsheet for the Ratio column uses the data in adjacent columns entitled “no. of Primes lower than next NonP” (Col.2) and “no. of NonP lower than this NonP” (Col.3) in order to calculate this ratio number.

The ratio is represented by a number to sixteen decimal places, seventeen decimal places are shown. Interestingly the ratio number naturally never exceeds sixteen decimal places.

The Number of Primes Less than ‘A Certain Number’

This observation relates to a previous observation, Observation 9 in PRIMES AND NON-PRIME PATTERNS; and this is now about the number of primes less than ‘a certain number’. The numbers of primes are shown in Col.2.

This certain number is the next non-prime odd-composite number, which is at the point of examination.

• The number of primes are shown in Col.2 as a continuing increase in value as the list is descended. The number is true to the rules of generation of this list of primes in accordance with my definition of a “prime number” developed strictly for this exercise.
• The list referred to, is my logical process list which produces primes and odd-composite non-primes. The odd-composite non-primes are only those numbers with prime factors equal to or greater than 7. Thus the primes; prime factors of 2,3, and 5; are excluded.
• This is the Set S_af.

In Col.3 is shown the number of non-prime odd-composite numbers below this row – the point of examination (i.e. not counting it).

For example, looking at row 34; Col.3 shows that there are 2 non-prime odd-composite numbers below this point, the point of examination, which is the row of the third non-prime odd-composite number 91. Indicated in Col.2, are 22 primes listed up to this point.

The Difference Number at random example Row 34.

The difference between Col.2 and Col.3 is indicated in Col.4, “DIFF of Prime and nonP”.

In this case the difference is 2 non-prime odd-composites minus 22 primes, i.e.-20.

The number is shown as -20 because there are a greater number of primes below this non-prime odd-composite number of 2. The primes exceed the non-prime odd-composites at this stage.

See the excerpt from the Ratio Spreadsheet above.

The Resultant Ratio Number

The resultant ratio number in Col.5 is calculated by dividing Col.3 by Col.2, only where the numbers in the spreadsheet columns are adjacent.

Looking down the list of numbers it can be seen that the value of this sixteen digit number increases quite quickly from 0.05…etc…… to 0.99…etc……. when the difference value is at -1 in row 1307 just before moving to zero.

—————-

The Difference Value 0 and Ratio Value 1

Then at row 1314, the difference value is 0, which causes the ratio to the attain the value of 1 (as a whole number “1”) . However there is a factor of variability to the values which cause the ratio to fluctuate up and down slightly.

The ratio value increases above 1 and then drops to -4. In row 1326 and 1328 it returns to 1 (i.e. the difference is 0 again). This also happens in rows 1360, 1372, 1374, and 1376. Then the difference value and consequently the ratio value starts to increase and stays above 0 and 1, respectively.

The Ratio Value 2

In row 58752 the value of the ratio first changes to above 2 for the first time at 2.00005107252298000. The next exact ratio of 2.00000000000000000 occurs in row 58757, then in rows 58766, 58826, 58832, 58919, 58922. After this point the ratio increases in values permanently above 2.

Ratio Convergence

I was tempted for minute there to think that this number was converging on a maximum of 2; so that it might be a converging series.

It cannot of course be converging towards a maximum of 2. If 1.9…+ occurred and next 2.0000000000000000, then it would represent a finite end point-value in a table that has to be infinite.

The ratio number relies on an infinite series Set S_af to provide the data for the calculation, so this ratio series must be infinite. Therefore there must be a difference between a prime and an odd-composite number forming a ratio number greater than 2. The series goes on past 2 as can be seen in the spreadsheet and is stated above.

I am going to create further spreadsheets to continue the charting of ratio numbers. See further posts which I will title “Complementarity – Ratio – etc. See names “Complementarity-Ratio-(no.)…” in the file names.

The Data in the Spreadsheet has been converted in to a Graph

The data in all the half million rows has been converted into a graph at the top of the sheet. This graph has been rotated clockwise by 90 degrees so that it displays vertically at the right hand side of the spreadsheet.

If one could expand the detail in this graph, it would show a very bumpy line. This bumpiness probably gives the relationship of the primes to non-primes the ability to be infinite, due this random aspect in the data.

There is this unpredictability causing the ratio not to be smooth. If it had been a smooth curve, there might have been a chance of convergence.

Proceeding up the Number-Line the amount of non-prime odd-composite numbers clumped together in groups will become larger and larger. The number of primes meanwhile will dwindle with the number of primes in groups becoming rarer and rarer.

Hence the proposal for further spreadsheets in order to see what form this information takes.

The Size of the Spreadsheet

I increased the number of rows in the spreadsheet from 50,000 to 500,000, specifically confirming that the ratio number passes the value of 2 for the first time at row 58752; the point of examination being non-prime odd-composite number 220283.

Below this level there are 19580 prime numbers (that obey my rules for formulating the primes list).

If a mathematicians prime definition is used then the numbers of primes would be increased by +2 through out the spreadsheet. This increase would occur by the addition of primes 2, 3, and 5, and the subtraction of 1 which I include as a prime number.

To build the spreadsheet, 20210218-PnonP-Ratio-orig20210116, which is 81Mb in size; the formulas, which are consistent in columns, had to be defined and named to reduce the file size and save LibreOffice from continually crashing.

Winge – There are things that are not ideal with LibreOffice Calc: –

• 1. There IS a glass ceiling on file size calculation, which must be dependent on the size and structure of formulas in the spreadsheet. This may or may not be influenced by the programs method of caching the development of the calculable file contents over time and also formatted content, in temporary files: there is a total of 32Gb RAM.

• 2. It is clear that the way the program is written, it cannot take advantage of a multicore chip (6 cores). Only one core is used to its full potential constantly (maxxed at 100%) whilst other cores are hardly used above 10%. This appears to happen on loading and as soon as a recalculation or save is required. I’m just asking this program to do too many complex calculations! While I have managed to construct the file – now when asking it to save additions or deletions and changes; it recalculates the entire sheet which creates a runaway on resources. Firstly the 32Gb of memory is used up and then the 64Gb swap file is used, with the LibreOffice crashing when the swap file reaches 100% usage.

• 3. Also, and really annoyingly; if you have a large spreadsheet loaded and you instigate a SAVE or an AUTO SAVE starts; this locks the whole of LibreOffice completely. So if you thought you might continue writing in Write whilst saving in Calc, you can’t; you just have to wait for 15 to 20 minutes while this large spreadsheet is recalculated and saved. The only way round this is not to use Write for composition of documents, but to use any other program that is a good text editor. This in itself is a problem, because the formatting of your composed documents then are differently based by the different program. They will (probably) change if you consequently load them in LibreOffice, due to problems with formatting.

• 4. All this is very annoying as it makes the concept of LibreOffice self defeating in its usefulness. Hey ho !