Author – Andrew J Frost 10/06/2020 REV TWO PnonP p8
OBSERVATION 8 Patterns:- Gaps – Other Places
Previously in Observation 7
The previously shown pattern of gaps, 6, 4, 2, 4, 2, 4, 6, 2, now turns up in other places.
Odd-composite numbers in my special Set Saf are in fact the product of two or more primes as prime factors. The value of the initial prime as prime factor being 7. The series starts with 7 x 7, 7 x 11, 7 x 13, and so on.
The principal in the spreadsheet table demonstrates the gaps between these numbers start at 42 followed by 28, 14, 28, 14, 28, 42, 14, 42. If we divide these gaps by 7 (the factor being considered), we get the first number as 6, with the following sequence 4,2,4,2,4,6,2 and this block pattern of eight numbers 6,4,2,4,2,4,6,2 is the same regularly repeating pattern as shown in the previous post.
- See also this previous post: – “Factors Patterns – Primes and Odd-Composite Gaps – 06/06/2020” (and also “Twin Primes and Ending Numbers“) for additional detail.
The Same Gap Pattern in a Different Context
The gaps series for the prime factor number 7 is analysed in blocks of 8 digits after the initial starting number 1 as follows: –
1,
6 4 2 4 2 4 6 2 , 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 2 4 6 2, 6 4 2 4 ¦ 2……
- ¦ = red boundary line on spreadsheet at integer 500.
- This red boundary line is where I stopped calculation of the prime and non-prime numbers and just used the pattern sequences I have discovered (two different ways) to continue inputting the prime and non-prime series of numbers into the spreadsheet table onward from integer 500.
- Note the prime and non-prime number list does not identify between primes and non-primes.
- I then used a list (by Alpatron) of all primes to input “P” to identify primes. I used a spreadsheet formula so that there were no errors to actually check-identify the primes themselves on down the list.
For prime factor 7: –
Factor 7, the sequence goes 49 – 7 = 42(6), 77 – 49 = 28(4), 91 – 77 = 14(2), 119 – 91 = 28(4), 133 – 119 = 14(2), 161 – 133 = 28(4), 203 – 161 = 42(6), 217 – 203 = 14(2)…….etc.
The Principal.
The first number is the prime under consideration (7) multiplied by factor 7. The result of this is an odd-composite number (49). The same prime number (7) is subtracted to give in this case (42). The resultant number (42) is then divided by the prime factor under consideration (7). This final result is the gap sequence number (6).
The Other Primes’, Prime Factors
The gaps series for the prime factor number 11 starts in the same way at 1, progressing into the same repeating pattern as above. There are less odd-composite numbers in the list for 11 up to the 500th integer at ¦ row 1605, as its value is greater.
All of the gaps series for the odd-composite numbers acting as prime factors in the Number Line, i.e 7, 11, 13, 17,19, 23 ….. and onwards, will follow the same sequence pattern. The gaps sequence is rigidly maintained.
The Principal of Gaps in the Prime Factor Sequencel
Taking this principal explained above, the same can be demonstrated for any prime factor sequence.
- The result is in bold type and the gaps are in brackets.
Factor 11, 77 – 11 = 66(6), 121 – 77 = 44(4), 143 – 121 = 22(2), 187 – 143 = 44(4), 209 – 187 = 22(2), etc. – divide the result by the factor being considered, 11, the same pattern repeats.
Factor 13, 91 – 13 = 78(6), 143 – 91 = 52(4), 169 – 143 = 26(2), 221 – 169 = 52(4), 247 – 221 = 26(2), etc. – divide the result by the factor being considered, 13, the same pattern repeats.
Factor 17, 119 – 17 = 102, 187 – 119 = 68, 169 – 143 = 26, 221 – 169 = 52, 247 – 221 = 26, etc. – divide the result by the factor being considered, 17, and the same gap pattern repeats.
The gap sequence still holds up to scrutiny and does not fail, as can be seen.
The Spreadsheets
The spreadsheet is in my MediaFire account in the folder “Spreadsheet for PNonP Patterns” at the following address. Please copy and paste the address into your browser. The file sizes of the spreadsheets are from – 32.55 Mb up to 122.67 Mb.
View the file with “Main” in its title.
You may download the spreadsheet(s) file from this address.
- Address – https://www.mediafire.com/folder/ztev7t9nh0xk3/Spreadsheet+for+PNonP+Patterns.
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