Author – Andrew J Frost 10/06/2020 REV TWO PnonP p1
PRIMES AND NON-PRIME PATTERNS
OBSERVATION 1 – Developing the Primes and Non-Primes List
My Logical Process Formula, 12n + {1,5,7,11} = p(p&np), is the formula which dictates that the integer “n” should be multiplied by twelve and then the product should be added to by 1, 5, 7 or 11. The result of this addition being four numbers that are either a prime or non-prime odd-composite numbers; odd-composite numbers are the product of two primes (if we ignore 1). Patterns are produced.
Looking at the spreadsheet : –
Black is a Prime. Magenta is a Power. Blue is an Odd-composite Non-prime.
Column 1 is blank, Column 2 is headed ‘intgr’ = integer = any whole number; Column 3 is headed ‘12 times’ and shows the base numbers; Column 4 is headed ‘+p’ and shows the additive sequence of 1, 5, 7, and 11, in a special sequence order: Column 5 indicates when squares, cubes and higher powers occur: Column 6 shows primes in rows with a Capital P; Column 7 headed ‘PRIME nonP’ shows primes in black and non-primes in light blue and powers in magenta; Columns 8 to 19 show the lowest prime factor ‘div1’ followed by the remainder ‘ans1’, followed by ‘div’ and ‘ans’ for the next 5 prime factors, i.e. for up to the first six prime factors of each non-prime number. Then after a another “PRIME nonP” column, in the following columns, the prime factors are reversed showing the largest prime factor first.
The Resultants of the Uplift numbers 1, 5, 7, 11 – are primes less than 12.
If n = 0 then the resultants are 0x12 = 0; then 0+1, = 1, 0+5 = 5, 0+7 = 7, 0+11 = 11.
If n = 1 the resultants are 1×12 = 12; then 12+1 = 13, 12+5 = 17, 12+7 = 19, 12+11 = 23.
If n = 2 the resultants are 2×12 = 24; then 24+1 = 25, 24+5 = 29, 24+7 = 31, 24+11 = 35.
Thus the first two non-primes 25 and 35 occur in the table.
If n = 3 the resultants are 3×12 = 36; then 36+1 = 37, 36+5 = 41, 36+7 = 43, 36+11 = 47.
If n = 4 the resultants are 4×12 = 48; then 48+1 = 49, 48+5 = 53, 48+7 = 55, 48+11 = 59.
So the third non-prime 49 and fourth 55, occur in the table, and so on.
List of the first 100 Non-Primes including numbers ending in 5 – from my special list
25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 385, 391, 395, 403, 407, 413, 415, 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 527, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 589, 595, 605, 611, 623, 625, 629, 635, 637, 649…
One can see that in the combination of primes with non-primes, the prime-factors of the non-primes are in a sequence where they cycle upward in value in a regulated way, ensuring that no logical non-prime and its prime-factor’s are missed in this infinite set of numbers.
Complementarity in Primes and Non-Primes
Briefly here, one of the reasons that I have looked into this area of mathematical analysis is that I believe the patterns developed here are evidence of deeper “layers” of pattern information. These deeper layers of relationship information in primes and non-primes are what we don’t understand – yet. This deeper information is not necessarily “fixed” and exists in a state of “complementarity”. This state of complementarity I believe allows for a more flexible, intuitive and imaginative interpretation from the broad picture of the results obtained.
The series of non-prime odd-composite numbers that consist of prime factors that themselves are primes only according to my definition above (i.e. not including 2, 3 and 5), expressed only once for each prime factor combination. The combination is not repeated, which demonstrates the importance of the limiting series 1,5,7,11. Whilst I cannot prove it, I am assuming that the primes that are included in the list occur without missing any prime number. This is in step with my special list and is an inclusive infinite series of all the prime numbers (from prime 7 onward).
Up to the prime at 6011 (the end of this initial sample of numbers) there is no sign of a prime or odd-composite number or odd-composite prime factor relevant to this set/series, being missed.
Conclusion
One concludes a relationship of the primes in the number line with the SubSet of primes when occurring as a factors of the non-prime odd-composite SubSet.
Thus:
So 7×7 is similarly followed by 7×11, 7×13 and so on. The sequence occurs in the order of the resultant increasing in value, so 7×11=77 and 19×7=133 comes before 13×11=143.
The last non-prime in my table was initially 500×12 = 6000 raised by 5 = 6005, which is divisible by prime 1201 as one of its prime-factors. 1201 is identified in row 403 as a prime.
The table shows the information about identifying the prime 1201 only 1/5th of the way into the table, well before it is needed for the calculation of the non-prime 6005, another 1602 rows ahead in row 2005.
As below: – 
