Author – Andrew J Frost 10/06/2020 REV TWO PnonP p12
OBSERVATION 12 – The Graph Shape Pattern
The Graph Shape
View this file stored in Mediafire in “Spreadsheets Old” – https://www.mediafire.com/file/dgvsfdkgqni3emr/20180929-non-P-by-formula-af.ods/file 20180929-non-P-by-formula-af.ods. Sheet: non-P by formula duplicates pattern_6. Positioned as sheet 3.
(Note) – I started to look again at the gaps (Today 05/10/2018) between the prime and odd-composite number series but in reference to Equation 2 and the pattern it produces on the spreadsheet. This resembles the ‘shape’ of the gravitational force of a planet from its centre out into infinite space or Gauss’s formula for the charge on a hollow sphere also produces the same shape of graph. (rotate by 90 degrees).
The above illustration taken from the spreadsheet is unfortunately not good having lost some formatting in the copying process.
The pattern is produced by running the Sandor Kristyan’s Equation 2, but also has added features.
As the table is extended to an ever increasing set of numbers (Set Saf), in the column where a = 0, more and more of these numbers become duplicated and triplicates in the cells to the east. The graph shape (“wave”) eastwards always stays similarly and equivalently the same shape as it grows bigger.
The Graph Shape 2nd Try
Below I have again illustrated the “wave” shape by creating another spreadsheet and reformatting it ‘conditionally’ and this image has been successfully created and has been copied over. The shape must be rotated in your minds-eye anti-clockwise by 90 degrees to attain the relationship I have in mind (see below).
View this file stored in Mediafire in “Spreadsheets Old” – https://www.mediafire.com/file/ninvz3x6qrjzrnn/20190331-nonDupl-IDprimes-DIAGS-redgrey.ods/file.
The result is a lot better at demonstrating the similarity of the shape obtained and the equations for the gravitational field from the centre of the planet to the surface and then the field from the surface into outer space. Reference the gravitational Shell Theorem.
Back To The Spreadsheet Data
For the primes shown in the column where a = 0, identified on a grey background. The rows extending from the prime numbers or the non-prime numbers; finish at the end of the row where squares or ‘complex squares’, are shown. These numbers, to a power, are completing the row forming the diagonal shape. Note, by complex square, I mean a number that is the sum of, say two (or more) true squares, such as 7×7 and 7×7 = 2401. Or 7×7 and 11×11 = 5929.
Non-Duplicates and Unique Numbers
If a conditional format to identify ‘non duplicates’ is run on the spreadsheet as illustrated above, only unique numbers are identified in cells with a grey background. Columns are grey and rows are grey with red horizontal borders. These rows of unique numbers identified, make horizontal lines that only line up with the primes in the furthest west column where a = 0 (grey background).
To quickly test this I copied the primes indicator ‘P’ in column 7 of – file: 20180929–12x-to34000-primes-af, sheet: P&nonP-2018-07-30. For the first 1000 entries.
The indicator ‘P’ corresponded with the position of the primes already identified in this sheet. This confirms that the duplicate/triplicate horizontal line pattern showing up, directly references the odd-composite numbers (clear background as rows).
The search for the non-duplicate numbers directly references the prime numbers (grey background as rows) in this infinite series, because the prime numbers themselves are unique and not duplicated or triplicated.
Duplicate Numbers
The clear background in the wave shape, indicates that in rows corresponding to the position of primes and non-primes are very often composed of duplicated numbers.
However further to the east of the spreadsheet beyond the wave-shape, the rows formed correspond with non-primes as a clear background and prime numbers as a grey background.
Thus the primes are identified byt this method without previous knowledge of their position in the series Set (Saf), in the column where a = 0 (when using Equation 2).
Special Sequence
This pattern has originated from the use of the special sequence of integers, which when a = 0 continues on to the next integer by the addition of the repeating series identified in the previous post.
The ‘Observation 11’ pattern in my previous post; i.e., 3.21212313; develops numbers as follows. So starting at 3, thus 0 + 3 = 3, the next integer column is the resultant of the addition of +2 = 5, and then + 1 = 6, and then +2 = 8, and then 9, 11, 14, 15, 18, and so on.
This infinite sequence of integers, 3, 5, 6, 8, 9, 11, 14, 15…, formed by the addition of the above gaps series, was first identified as the gaps pattern 42424626, and is half the value of this gaps pattern.
Algorithms
- Note here that I am using the computer algorithms in LibreOffice Calc, to identify duplicates and non-duplicates by ‘conditional formatting’. It is these algorithms that are doing the identification of the primes in the existing table formed from Equation 2.
LibreOffice Calc being a Linux program, the actual code that runs to identify duplicates is probably accessible and editable, which would mean that in theory, a section of code could be used as a ‘source’ in order to identify primes specifically in Calc.
Prime Identification Algorithms
Primes however are not individually identified, it is just a collective property that they have when presented in a “map”, i.e. spreadsheet, with the condition of ‘unique numbers’ as clarified above.
It may be possible to rewrite the code that identifies unique numbers in Calc, in a way that works “outside” a spreadsheet. In other words a conceptual spreadsheet is run without visualisation slowing it down, in which the series of numbers in (Set Saf) is identified and the primes are identified in this because they are unique. Hopefully this would be quick!
On-Line Encyclopedia of Integer Sequences
OEIS – http://oeis.org/ The On-Line Encyclopedia of Integer Sequences.
- Note: -2018-11-18 Found and joined OEIS https://oeis.org
The On-Line Encyclopedia of Integer Sequences® (OEIS®)
The OEIS has records of the sequences of integers that I have discovered for myself by using my own logical process and then later by using the Equation 2 of Sandor Kristyan.
Below is a sequence showing odd-composite numbers (not including prime numbers – see spreadsheet) that are originated in my list, Set ‘S(af)‘. These numbers in A038510 are the complementary, odd-composite, non-prime numbers identified in my list.
OEIS Sequences
A038510 Composite numbers with smallest prime factor >= 7.
49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 343, 361, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 529, 533, 539, 551, 553, 559, 581, 583—
Therefore my list ‘S(af)‘ minus resultants of sequence A038510 = a list of primes.
A070884 7 + x where x is congruent to {0, 4, 6, 10, 12, 16, 22, 24} mod 30
7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 221
Therefore A070884 minus resultants of A038510 also = a list of primes.
Looking at the title of A070884, one can see the gap series I have identified occurs again here, i.e. 0, 4, 6, 10, 12, 16, 22, 24, mod 30 – as a series separated thus – 4, 2, 4, 2, 4, 6, 2, 6, which is the gap sequence occurring repetitively through my Set ‘S(af)’.
The following numbers developed from the named OEIS sequences, list the required integers in Equation 2 for a=0 & b=integer, to produce the complete set of values shown in the spreadsheet.
A005097 (Odd primes – 1)/2.
also
A130290 Number of non-zero quadratic residues modulo the n-th prime.
3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156
My spreadsheet uses and produces this eight number sequence which increases by +15, when and if the odd-composite complementary series has been removed. | marks the boundary of the repeating originating sequence 2,1,2,1,2,3,1,3, with the resultants as shown below. My “Intcol Integer Sequence”.
3, 5, 6, 8, 9, 11, 14, 15, | 18, 20, 21, 23, 26, 29, 30, 33, | 35, 36, 39, 41, 44, 48, 50, 51, | 53, 54, 56, 63, 65, 68, 69, 74, | 75, 78, 81, 83, 86, 89, 90, 95, | 96, 98, 99, 105, 111, 113 114, 116, | 119, 120, 125, 128, 131, 134, 135, 138, | 140, 141, 146, 153, 155, 156 ……
However yet more patterns are buried in these series originated in (Set Saf), also developing from Equation 2. Will look into this further in next and other posts.
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