Author Part 2 – Andrew J Frost 02/02/2022 PnonP p4
Groupings & Patterns Forming Structure – Andy Frost 2022/02/02
Structures?
Primes With a Purpose
Structure in the organisation of Prime numbers might be seen as a contentious subject title. However this is where I’m going with this page.
It is clear within my last post that there are definable patterns to primes when they are functioning as prime factors. There may be other patterns around the list of prime numbers, when they have another function not yet defined.
So the first “purpose” is, primes as factors.
In the previous page I showed that there are groupings of primes as factors; but only four groups; a, b, c, d. Every prime number is a member of one of these groups, but only one. The group is dependent on the ending-number of the prime or odd-composite-non-prime being considered.
As above; in the spreadsheet: – https://www.mediafire.com/file/299ywfx6o7aignp/aP%2526nonP_P-Factors-20220124.ods/file the data is arranged to show the relationship between the prime-factors and their products which generate the list of odd-composite-non-primes. The prime factors belong to one of the four identified groups.
Also shown, is the subSet of numbers where the product of the prime-factors creates a member of this subSet, which are the odd-composite-non-primes.
(To access/load some of my larger spreadsheets (128 MB) you will need to own a computer/laptop with 32 Gig. of memory and a good processor)
Previously Posted
Please see previous post “PRIMES AND NON-PRIME PATTERNS -12 – The Graph Shape” in which the sequence (subSet) below is shown.
OEIS Sequence A038510 – Composite numbers with smallest prime factor >= 7.
(subSet of odd-composites within my list SetS(af)).
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—
The Groups
Names
Taking the primes as factors (see the column), and the row labelled “factors”; it is clear from the spreadsheet that the groups are prime factor numbers ending in 1, 3, 7, and 9, and the pattern of these numbers are determined by the gap sequence 6,4,2,4,2,4,6,2 in the vertical column (limited to the first sequence**) and horizontal row as shown.
The groups as named are: –
- Factors ending in 1 are group “c”.
- Factors ending in 3 are group “a”.
- Factors ending in 7 are group “d”.
- Factors ending in 9 are group “b”.
The Repeating 8 Number Sequence (extracted from the product numbers).
| 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | ||||
| 7 | 9 | 7 | 1 | 9 | 3 | 1 | 3 | 7 | d | ||
| 11 | 7 | 1 | 3 | 7 | 9 | 3 | 9 | 1 | c | ||
| 13 | 1 | 3 | 9 | 1 | 7 | 9 | 7 | 3 | a | ||
| 17 | 9 | 7 | 1 | 9 | 3 | 1 | 3 | 7 | d | ||
| 19 | 3 | 9 | 7 | 3 | 1 | 7 | 1 | 9 | b | ||
| 23 | 1 | 3 | 9 | 1 | 7 | 9 | 7 | 3 | a | ||
| 29 | 3 | 9 | 7 | 3 | 1 | 7 | 1 | 9 | b | ||
| 31 | 7 | 1 | 3 | 7 | 9 | 3 | 9 | 1 | c |
- (** The first sequence follows the gap pattern, however this does not necessarily apply to following sequences – but the gap pattern evident between these group sequences is consistent and not random – gap patterns are infinitely repeating sequences – see next post no.24)
The Product of the Prime Factors
The table above shows the ending number of each of the products of prime factors. The sequence of ending numbers is dependent on the first product of factors. The first prime factor is of course 7. The product is 49 – ending number 9. The first sequence of numbers forming a group is “d” therefore.
Group “d” and other groups “a, b, c”.
Group “d” which is equivalent to this sequence 9, 7, 1, 9, 3, 1, 3, 7. This is a unique sequence which repeats, as do each of the other group sequences. Every eight ending numbers, related to their prime factor product numbers.
The numbers shown are the ending numbers of the product of two prime-factors. These are the numbers for group “d”. These are the eight ending numbers that repeat indefinitely for this group. This principal applies to groups a, b, and c.
Primes
The First Few Rows
Such is the nature of the numbers in the list Set S(af) that the first 13 rows in the spreadsheet are all primes.
The First Odd-Composite Non-Prime
The fourteenth is the first odd-composite-non-prime, 49. Not able to act as a prime factor because it is the product of prime factors. Whilst the numbers are not shown in the row on the spreadsheet, it is easy to work these out for yourself, or scroll sideways to the rows and columns controlled by the polynomial formula for Eq2, where they are shown.
-
- if you are not aware of equation Eq2, please see my previous post: –
- https://andyfrostglassandart.eu/wp-admin/post.php?post=869&action=edit
It is clear they fit in to same pattern for the ending number. In the case of 49 it is (49 x 49 = 343) which means that the pattern of eight ending numbers starts with 3.
The Gap Pattern Vertically
Looking at the vertical application of the of the group names/numbers; the formula for the application of the gap pattern for the product of primes as factors is different as previously explained.
-
- if you are not aware of “P&NonP – 20 – Positions of Prime Factor Patterns in Odd-Composites pg1” please see my previous post link below: –
- https://andyfrostglassandart.eu/wp-admin/post.php?post=1256&action=edit
- So the formula for this calculation is: – (prime factor1 x prime factor2 = non-prime odd-composite) – (previous odd-composite)/prime factor = resultant gap number)
- Please read the whole post to understand the origin of the information
You can see the physical spreadsheet representation of this in the link specified above for the spreadsheet, saved in my MediaFire account.
The area of the spreadsheet concerned is to the left of the information about “grouping”, discussed above.
Visualising the Information
Sheet Structure
The information could be considered as several overlapping sheets, each describing a pattern relationship between the gap pattern 6, 4, 2, 4, 2, 4, 6, 2; – the numbers being primes and odd-composite non-primes,
By looking at the numbers (or their ending numbers, or the gaps between them) in the list Set S(af) or the two complementary subSets; i.e. contained Set Sc1 & Sc2; we may be able to begin to visualise the x,y sheets of data with the way these “sheets” of information relate to each other.
Sheets would be linked by common elements of data and common links of meta-data patterns.
See link below to the post about the Ratio.
https://andyfrostglassandart.eu/complementarity-ratio-3/patterns/2021/06/02/
So – what next?
Well the above is a start to illustrating the idea of Structure, even if it is purely “cosmetic”. (Or mathematically “combinatorics”.)
All the posts of pages right from the one labelled 00 in its title; the starting page/post; these are really to show that there is some form of structure behind these numbers.
I wrote these pages as a 20 page paper in 2018 and tried to publish, all be it unsuccessfully. Now I have added additional pages and side issues to get to this point.
Not Random
Structure is possibly ‘weird’ as ideas go for something as irregular and random as prime numbers. Structure implies regularity and as a consequence, the idea that prime numbers or their complementary numbers may not actually be random at all.
Surely too far fetched? Not really; because I believe that what links these numbers is a “state” considered as a concept. Imagine a situation rather like the physical states of matter where a phase boundary is crossed, going from gas to liquid, liquid to solid etc. It is this boundary and the process of “crossing” that needs elucidation.
There is also the idea that while prime numbers have a random distribution, they might actually approximately be traceable to their immutable positions in some definable way by using the odd-composite non-prime sequences.
end
Appendices
Rough Notes
Prime – eight ending numbers which are where this prime occurs as a factor in an odd-composite named in sequence as well
prime
– sequence of ending numbers
– location in odd-composite
–7 – 9719 3137 49,77,91,119,133,161,203,217…
6 4 2 4 2 4 6 2 6
-11 – 7137 9391 77,121,143,187,209,253,319,341…
6 4 2 4 2 4 6 2 6
-13 – 1391 7973 91,143,169,221,247,299,377,403…
6 4 2 4 2 4 6 2 6
-17 – 9719 3137 119,187,221,289,323,391,493,527…
-19 – 3973 1719 133,209,247,323,361,437,551,589…
-23 – 1391 7973 161,253,299,391,437,529,667,713…
-29 – 3973 1719 203,319,377,493,551,667,841,899…
-31 – 7137 9391 217,341,403,527,589,713,899,961…
-37 – 9719 3137 259,407,481,629,703,851,1073,1147…
-41 – 7137 9391 287,451,533,697,779,943,1189,1271…
-43 – 1391 7973 301,473,559,731,817,989,1247,1333…
-47 – 9719 3137 329,517,611,799,893,1081,1363,1457…
-53 – 1391 7973 371,583,689,901,1007,1219,1537,1643…
-59 – 3973 1719 413,649,767,1003,1121,1357,1711,1829…
-61 – 7137 9391 427,671,793,1037,1159,1403,1769,1891…
-67 – 9719 3137 469,737,871,1139,1273,1541,1943,2077…
-71 – 7137 9391 497,781,923,1207,1349,1633,2059,2201…
-73 – 1391 7973 511,803,949,1241,1387,1679,1903,2117…
-79 – 3973 1719 533,869,1027,1343,1501,1817,2291,2449…
-83 – 1391 7973 581,913,1079,1411,1577,1909,2407,2573…
-89 – 3973 1719 623,979,1157,1513,1691,2047,2581,2759…
-97 – 9719 3137 679,1067,1261,1649,1843,2231,2813,3007…
101 – 7137 9391 707,1111,1313,1717,1919,2323,2929,3131…
103 – 1391 7973 721,1133,1339,1751,1957,2369,2987,3193…
107 – 9719 3137 749,1177,1391,1819,2033,2461,3103,3317…
109 – 3973 1719 763,1199,1417,1853,2071,2507,3161,3379…
113 – 1391 7973 791,1243,1469,1921,2147,2599,3277,3503…
127 – 9719 3137 889,1397,1651,2159,2413,2921,3683,3937…
Above is a repeating sequence of eight ending numbers in certain odd-composites in the list Set(af) where each prime number is occurring as a factor.
Below are sequence matches for the prime factors concerned: –
-07 matches 17, 37, 47, 67, 97, 107, 127…
-11 matches 31, 41, 61, 71, 101…
-13 matches 23, 43, 53, 73, 83, 103, 113…
-19 matches 29, 59, 79, 89, 109…
it is clear that:-
1) the sequence infinitely repeats after the first eight ending numbers
2) the eight digit sequence is likewise grouped into only four combinations
3) the prime factors are grouped into primes ending in 1, 3, 7, and 9, i.e. four groups, according to the four sequences, infinitely
