Author Part 2 – Andrew J Frost 08/02/2022 PnonP p5
Patterns Forming Structure – Andy Frost commenced 2023/03/26
(wow – the date now is 19/2/2024)
Review of Structure
First a reprise of former conclusions – (apologies for repetition but I feel it is important to clarify previously noted crucial patterns and their “positions”). Also since my last post in April 2022 a lot of time has gone by and I have changed the name of this site. Previously it started life as “15711.org” then changed to “if-prime.org” then “andyfrostglassandart.eu” and now andyfrost.xyz.
This post is the first made with my new site name. There will inevitably be teething problems.
To continue: –
If you are reading this post and do not know to what the items below refer to, please look at my previous posts first and also the spreadsheets in my Mediafire account. Links to my Mediafire account are shown in my previous posts. Here’s a link to a “how to” document to take you to my Mediafire: –
https://www.mediafire.com/file/r09r1ry3860o8r5/MediaFire_-_Getting_Started.pdf/file
Structure of Primes and Prime Factors and Odd-Composites
The crucial element is the established gap pattern of 6, 4, 2, 4, 2, 4, 6, 2 – eight numbers representing the value of gaps which occur repeatedly in “infinite” sequences. This gap number sequence is always maintained but does not necessarily start with the same number/place in the sequence. This starting number in the sequence occurs in different ways in different series of numbers and very often is 2, 6, 4, 2, 4, 2, 4, 6.
The initial logic of the explanation of these elements below is my process equation for listing the prime numbers and specific odd-composite numbers –
My logical process formula for the generation of my special Set S(af) of prime and non-prime odd-composite numbers: –
┌ 1
12n + ├ 5 = p(p&np)
├ 7
└ 11
where: – p(p&np) – [div5] = S(af) “p(p&np) minus [div5]”
where: – n is an integer
1,5,7,and 11 are prime numbers (uplift numbers) used to raise the resultant of 12n to “p(p&np)”
p(p&np) are primes and non-primes evolved from odd-composite numbers
[div5] is redundant information (i.e. all odd-composite numbers ending in 5)
hence the ‘minus’ which represents a sieve (or sort) to remove them.
S(af) my Set comprises all the primes and all the non-primes composed from odd-composite numbers.
Patterns
1) – Controlling Set – 1,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0. Numbers which control the 16 digit change pattern of the integer numbers. Each time 1 occurs the determining integer series is uplifted by this value i.e. integer 1 becomes 2, integer 2 becomes 3 and so on… So the first integer is used three times then uplifted by 1, the second integer is used four times then uplifted by 1, the third integer is used twice, the fourth integer is used four times and finally the fifth integer is used three times… The sequence then repeats. The sequence in numbers is one hundred, one thousand, ten, one thousand, one hundred.
2) – Special Set Saf – Gap Sequence between the numbers of my special Set Saf, regulated by the sequence 6, 4, 2, 4, 2, 4, 6, 2
This is the gap between all the numbers in the sequence, primes and odd-composites =>7.
3) – Ending Numbers – the Gap Sequence between the ending numbers of all numbers in the list Set Saf .
The pattern within the list is completely rigid, 1, 7, 1, 3, 7, 9, 3, 9, 1 (8 numbers) as regulated by 6, 4, 2, 4, 2, 4 6, 2.
3a) – Ending Numbers – the Gap Sequence between the ending numbers of all numbers in the list Set Saf .
The pattern within the list can be express by the spreadsheet formula =MOD(RC[-1]-R[-1]C[-1],10). The “10” representing a clock sequence based on a digital clock count of 10.
4) – Uplift Numbers – the uplift number sequence – 1, 7, 11, 1, 5, 7, 11, 5, 7, 1, 5, 7, 11, 1, 5, 11, (16 numbers continually repeating sequence) as regulated by 6, 4, 2, 4, 2, 4 6, 2.
4a) – Uplift Numbers – the Gap Sequence between the uplift sequence of all numbers in the list Set Saf .
The pattern within the list can be express by the spreadsheet formula =MOD(RC[-2]-R[-1]C[-2],12). The “12” representing a clock sequence based on a standard clock count of 12.
5) – Integers sequence of Equation 2 – (0) 3, 5, 6, 8, 9, 11, 14, 15, then 18, 20, 21, 23, 24, 26, 29, 30 and so on.
The Gap Sequence between these integers is the same, except that the values are halved compared with the previous gap sequence, i.e. the sequence actually starts at 3,2,1,2,1,2,3,1 as half of 6, 4, 2, 4, 2, 4 6, 2.
6) – Specific Odd-Composites. The Gap Sequence of primes when the primes occur as factors of odd-composite numbers specifically in the list. Ref: post P&NonP – 20 (page 1). The controlling sequence as explained in the previous post, is 6, 4, 2, 4, 2, 4 6, 2. Taking prime 7 as an example, we cycle through the odd composites 49, 77, 91, 119, etc., then the same for prime 11, 13, 17, and so on.
7) – Prime Uplift – the number required to create the list Set Saf from the initial integer-times-12 even-number, plus 1, 5, 7, 11.
Pattern 1: – 1, 7, 11, 1, 5, 7, 11, 5, 7, 1, 5, 7, 11, 1, 5, 11, (16 numbers) as regulated by 6, 4, 2, 4, 2, 4 6, 2.
The spreadsheet formula for the sequence is =MOD(RC[-1]-R[-1]C[-1],12) and derives the sequence from the base of mod 12.
8) – Factors of Specific Odd-Composites. The Gap Sequence of primes when the primes occur as factors of odd-composite numbers specifically in the list. Ref: post P&NonP – 20 (page 1). The controlling sequence as explained in the previous post, is 6, 4, 2, 4, 2, 4 6, 2. Taking prime 7 as an example, we cycle through the odd composites 49, 77, 91, 119, etc., then for prime 11, 13, 17, and so on. Note the black emboldened numbers are the “ending” numbers of the product of the primes, the odd-composite (prime x prime) number is shown in the next line below. Red numbers are squares.
Factors of Specific Odd-Composites.
| prime | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |||
| 7 | 9 | 7 | 1 | 9 | 3 | 1 | 3 | 7 | d | ||
| 49 | 77 | 91 | 119 | 133 | 161 | 203 | 217 | ||||
| 11 | 7 | 1 | 3 | 7 | 9 | 3 | 9 | 1 | c | ||
| 77 | 121 | 143 | 187 | 209 | 253 | 319 | 341 | ||||
| 13 | 1 | 3 | 9 | 1 | 7 | 9 | 7 | 3 | a | ||
| 91 | 143 | 169 | 221 | 247 | 299 | 377 | 403 | ||||
| 17 | 9 | 7 | 1 | 9 | 3 | 1 | 3 | 7 | d | ||
| 119 | 187 | 221 | 289 | 323 | 391 | 493 | 527 | ||||
| 19 | 3 | 9 | 7 | 3 | 1 | 7 | 1 | 9 | b | ||
| 133 | 209 | 247 | 323 | 361 | 437 | 551 | 589 | ||||
| 23 | 1 | 3 | 9 | 1 | 7 | 9 | 7 | 3 | a | ||
| 161 | 253 | 299 | 391 | 437 | 529 | 667 | 713 | ||||
| 29 | 3 | 9 | 7 | 3 | 1 | 7 | 1 | 9 | b | ||
| 203 | 319 | 377 | 493 | 551 | 667 | 841 | 899 | ||||
| 31 | 7 | 1 | 3 | 7 | 9 | 3 | 9 | 1 | c | ||
| 217 | 341 | 403 | 527 | 589 | 713 | 899 | 961 |
It is clear that the ending numbers produce 4 distinct sequences as denoted by the letters a,b,c,d – cf below. Also the 4 sequences produced are beginning and ending – 9719…3137, 7137…9391, 1391…7973, 3973…1719,
Excerpt from Rough Notes below:-
–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…
——————————-6—4—2—-4—-2—4—-6—-2—-6
-19—- 3973 1719—-19, 29, 59, 79, 89, 109…
——————————-6—4—2—-4—-2—4—-6—-2—-6
The first eight digits corresponding with the prime number is the sequence of ending numbers for that group. The numbers in the group always correspond to the immutable pattern of eight digits 6, 4, 2, 4, 2, 4, 6, 2, which is the Gap pattern.
-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…
The first five lines are showing the Gap pattern that occurs between the numbers of each type/group. the sequence is identified by an eight digit sequence of ending numbers of which there are only four combinations. Eg. the number 7 and 17 have the same same sequence. All sequences adhere to the Gap pattern.
To continue the grouping of 7, 17, we can see the next is 37, 47, 67, 97, 107, and 127 and so on.
The number 11 is the start of the sequence 11, 31, 41, 61, 71, 101…
The number 13 is the start of the sequence 13, 23, 43, 53, 73, 83, 103, 113…
The number 17 is the start of the sequence 17, 37, 47, 67, 97, 107, 127…
The number 19 is the start of the sequence 19, 29, 59, 79, 89, 109…
The number 23 is the start of the sequence (13), 23, 43, 53, 73, 83, 103, 113… i.e. the sequence is the same as 13 but obviously starting at 23, which happens with all the primes onward. These groups are prime numbers ending in 1, 3, 7, 9.
The prime numbers, no matter what their numerical size are members of one of these four groups, of which the next number in the group is determined by the Gap pattern. The Gap pattern being an eight digit sequence has to, itself be in the correct correspondence with the ending number sequence. Therefore there is the possibility to determine the the next potential prime by application of these patterns.
