Author – Andrew J Frost 10/06/2020 REV TWO PnonP p6
OBSERVATION 6 – Patterns:- Squares and Powers
In the spreadsheet table, odd-composite numbers that are squares or other powers are shown as a magenta coloured font.
- See the copy of part of the spreadsheet shown in a previous post or at the following address. Previous post -03 – https://15711.org/patterns-3/patterns/2020/12/14/
The Spreadsheets
The spreadsheet is in my MediaFire account in the folder “Spreadsheet for PNonP Patterns” at the following address. Please copy and paste into your browser. The file sizes of the spreadsheets are – 32.55 Mb and 122.67 Mb.
You may download the spreadsheet file from this address: – https://www.mediafire.com/file/s5tof7b0edxpd38/20210101-PnonP-Main-orig20191130-Main.ods/file
Please let me know if you have a problem accessing this or any other spreadsheet.
Review Beginning
To review the beginning of this discussion on Primes and odd-composite numbers and understand the train of thought of this part of my blog, please reference this post: – https://15711.org/primes/patterns/2020/11/22/
Even Powers
The occurrences of non-prime odd-composite numbers that are squares, only happen for numbers raised from the integer series list of ‘times 12’, by the special integer 1.
Powers of 2 and then powers 4,6,8…and so on, are even exponents. They only occur in this sector of the initial integers where they are “raised by” 1.
Odd Powers
The exception to the above “even” rule are the exponents that are odd numbered powers.
All the odd-composite numbers with a factor subject to odd numbered powers 3, 5, 7, 9, and so on; can occur in any row “raised by” any of the special integers 1,5,7,11.
These series of non-prime odd-composite numbers logically follow on ad infinitum. No sequence of odd-composite numbers to a power are missed in the logical development of the prime and non-prime series in the table.
This becomes evident in the spreadsheet when an equation is used to develop the numbers shown, currently developed in, but in addition to my special Set S(af) – list; as will be explained later.
My logical process method for the generation of my special Set S(af) comprising prime and non-prime odd-composite numbers: –
12n + {1,5,7,11} = p(p&np)
and p(p&np) –[div5] = S(af)
where: –
- n is an integer – formed from a special integer sequence
- 1,5,7,and 11 are prime numbers used to raise the resultant of 12n to “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 (subSet ‘primes’) and the non- primes (subSet ‘non-primes’), composed from odd-composite numbers according to the origination process above.
