10/06/2020 REV TWO, published 22/11/2020.
PRIMES AND NON-PRIME PATTERNS
This is about my interest in primes as an expression of patterns or sequences of patterns.
This has happened as a progression of personal discoveries. These documents deal with basic maths; there are no advanced forms of maths in them, because such is outside the scope of my knowledge. This is about visual number patterns, which give greater insight. These are ‘relationship patterns’ within Sets of numbers.
While searching for numeric patterns and then consequently having discovered such patterns; I also noticed initially that I could not find references to these patterns or that anyone had seemed to have noticed them before. If they were on record, no observations had been drawn. Later I found OEIS.
Some, not all, of these patterns in numbers are purely cosmetic; that is they appear to serve no significant purpose. However I believe others are far more significant. Even if a pattern is cosmetic, it can contribute to the overall picture of what is going on.
Of course, because I am talking about primes; in the back of my mind was the thought of being able to identify them without the need for proving the number as a prime. The normal way is to work out an individual elimination-calculation by using prime factors to identify those numbers that are not prime (non-prime) or prime; one number at a time. To say the least, laborious.
A prime number is any number that can only be divided by itself once; that is, its prime factors are itself and the number 1. A non-prime therefore, is any number with more than two prime factors** which are greater than 1.
** Prime factors are numbers that are prime; i.e. 2, 3, 5, 7, 11, 13, 17, 19, 23… and so on. These are numbers, that when multiplied together give another number that is not prime.
Curiosity got the better of me.
I first started looking for patterns in the prime numbers series, in March 2018. The reason for my research in primes stemmed from a completely different enquiry that I was engaged with, which I currently continue as a separate project.
With the primes, I started by scribbling number series on lose pieces of paper as you do, as a consequence, was intrigued by the idea of using a base number 12, as I explain below.
On the internet the most common way of expressing primes is that they can occur every 6 integers (numbers), plus or minus 1.
No negatives
At the time I did not want negative numbers involved in my work to determine a prime. It seemed to me that the best way of getting round this was to use any number, call it an integrer ‘n’, and multiply it by 12. Then to raise the resultant by the addition of one of a selection of four positive primes that occur below 12.
That is 1,5,7,11.
For this exercise I regard the number 1 as a true-prime. Indeed how can there be a prime number without it being divisible by itself just once? However I do note that considering the number 1 to be prime causes problems elsewhere in mathematics, also as it can be shown not to be “prime”.
I did not include 2 as it is not, in my mind a true-prime, because it is an even number and the only even numbered prime, therefore it cannot factorise an odd-composite *N1* number into whole numbers. Nor did I include 3, as in this exercise it produces a list of resultant odd-composite numbers that are guaranteed not to include any prime numbers; all resultants being divisible by 3 and this conclusion is equally true for 5; all resultants being divisible by 5, therefore not prime.
*N1* Odd-composite numbers, in the context of this document are whole odd numbers that can be formed from the multiplication of two or more prime numbers; this multiplier rule of course excludes 2, 3 or 5, as explained above.
I developed my list/table in spreadsheets which use the addition of the prime numbers 1,5,7, and 11 in the manner described below. The list includes prime numbers and odd-composite numbers, which I sometimes call “non-prime” numbers (see Note below).
*N2* The prime factors of the non-prime odd-composite numbers in this special list are all primes that always previously occur in the list.
The list (my special list) starts off with an uninterrupted, infinite series of primes and non-primes in this special prime series: – 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49….. -(any numbers ending in 5, including 5 itself, can be omitted by sieving out, which is easy to do in a spreadsheet ‘sort’).
Note – the expression “non-prime” I am using, is to identify a specific and limited list of odd-composite numbers. These odd-composite numbers in this list (eg. red numbers above) are always the product of more than two prime factors, which once numbers ending in 5 are sieved out are then factors as odd primes greater than or equal to 7.
My logical process formula for the generation of my special list of prime and non-prime numbers (including 5): –
12n + {1,5,7,11} = p(p&np)
where: –
n -is an integer (whole number)
1,5,7,and 11 -are prime numbers used to raise the resultant of 12n to “p(p&np)”
p(p&np) -are primes and non-primes -the non-primes are evolved from odd-composite factorial prime numbers and are all odd-composite numbers themselves; i.e. part of my special list.
