Author – Andrew J Frost 29/08/2021 REV TWO PnonP p15
OBSERVATION – 15 – Prime Positions
Identifying Prime Positions – (it is Just an Idea)
Going back to Observation 9. Today (23/02/2019) I put together a new spreadsheet to check out if primes in Set Saf were actually being identified. This is not an exercise in showing the distribution of primes as the Riemann Conjecture, but to independently identify primes without actually calculating them.
By laboriously doing the sums of division required to try and find whether or not there are prime factors of the numbers under examination, if not the number is possibly a prime.
Therefore the process required would be to write an executable code that runs the various steps. Unfortunately there is no chance that I would write this code as I have no knowledge in this area, but below are the considerations one might require. The code would non-visually create and run the following steps; the last step being the concept of spreadsheet conditional formatting in code.
My Set.
Having established my set of numbers Set (S(af)) and then found Sandor Kristyan’s paper which includes a polynomial Equation 2; use Equation 2 where a = the integer sequence (Intcol) and b = 0.
The Gap Sequence.
Use the Gap Sequence with the Integer Pattern; i.e. the integers horizontally and vertically, named as ranges “introw” and “intcol”, that are spaced according to the following integer gap pattern: – 2,1,2,1,2,3,1,3, which then repeats. This pattern is the initially identified pattern 4,2,4,2,4,6,2,6 divided by 2; the pattern 4,2,4,2,4,6,2,6 defines the Set (S(af)).
The Integer Pattern (“Intcol Integer Sequence”).
The “introw” and “intcol” sequences are actually ranges of integers of the values: –
3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 24, 26, 29, 30, 33, ….. for both horizontal and vertical streams of integers. Each 8 number line of integers in the pattern is increasing in value by +15 as a repeating sequence (i.e. 18 is 3 + 15, 5 + 15 = 20, and so on) but we are only interested in the column where b = 0.
Use The Prime Identifier Column.
Use the prime identifier column (previously constructed list or any published list), as a way of checking that primes are correctly identified by the method immediately below by “Conditional Format”.
In The Spreadsheet – The Conditional Format.
Apply a conditional format to the resultant of the Calc formula initially to identify duplicate numbers, non-primes, so checking for identification errors.
There are no failures in identifying non-primes in the list and as a consequence, identifying the primes, because they are unique numbers. As a prime is a unique number, it originates an entire row of unique numbers once it is outside the graph wave shape field (please read previous posts defining this), therefore the row receives no coloured background. (NB. In Calc it is possible to identify duplicates and/or non-duplicates, i.e. prime numbers would show up formatted to a colour, which is what happens; thus producing the graph wave shape.)
Errors
However, in this checking spreadsheet (see links to “test” spreadsheets below), move to row 3406 and proceed to check the rows and numbers backward up the sheet and errors are identified. (These errors are shown by a black ‘F’ for failure in the prime identification column where I have check-identified primes shown by a red ‘P’).
This indicates that conditional formatting has to be increased by further extending it to more columns in a proportion eastward (proportion not calculated); also to increasing the number of rows (much greater) in the conditional format in the extended sheet.
On addition of a further conditional format indicated by a blue colour, the rows that had failures can be seen then to be identified as primes by blue areas corresponding to the non-primes in the list. This action has therefore minimised or possibly cleared the potential for errors.
Non-duplicates
Following this I have reproduced the sheet using non-duplicates in the conditional format.
Therefore to summarise, as columns are added to the east of the spreadsheet, rows must be added going south in a much larger proportion. This proportion is not a constant and has to be an increasing function. The increasing ratio between columns and rows is not identified in order to maintain the relationship that creates the graph shape identified in Observation 12 and Appendix 6.
2401: columns 14, rows 646, i.e. 46 times 14 columns approx.
5929: columns 21, rows 1587, i.e. 75 times 21 columns approx.
10609: columns 28, rows 2835, i.e. 101 times 28 columns approx.
The ratio itself of the increase in number of rows southward, is decreasing relative to itself.
Number Crunching Calculations of Primes
Using this idea in an unlimited way would lead to an information overload unfortunately; quite quickly. The calculation constantly expands and becomes too big for any computer to handle. For this sort of coding to be of any use, limits would have to be set. A starting number from the Number Line would have to be selected, then an ending limit number for the calculation; for example, say 500,000 rows. There would have to a starting number which could either be a known prime, or could be a known odd-composite non-prime from the list, i.e. with factors that are prime factors (these occur in a unique combination for each odd-composite-non-prime).
What is actually being identified is a group of results (i.e. a Set of numbers), with primes being highlighted because primes are unique numbers. There is not however a defined identification of each unique number, and (as usual) a runaway calculation would occur.
Test spreadsheets are here: – https://www.mediafire.com/file/ap4og0tjgmkcbmd/20190331-nonDupl-IDprimes-DIAGS-redgrey.ods/file and https://www.mediafire.com/file/x61qvpeojdsmowq/20190222-pnonp_setAf_-c3-purplebluish.ods/file
