Author – Andrew J Frost 10/06/2020 REV TWO PnonP p9
OBSERVATION 9 Pattern:- Gaps – Difference between Primes & Non-Primes
This observation is about the number of primes less than a certain number and the number of non-prime odd-composite numbers less than the same number; describing the difference between the two values.
A somewhat dryer than average subject.
Column headers for this analysis are as follows: –
Below is included a table form of the spreadsheet after the description of the column headers which is the same for the spreadsheet image as well as the table below.
- First titled column shows the prime numbers.
- Second titled column shows the non-prime odd-composite numbers.
- Third titled column shows both the prime and non-prime numbers.
- Forth column shows the gaps sequence between prime and non-prime.
- Fifth column ‘pattern 4’, ending digit of the integers.
- Sixth column ‘pattern 5’, ending digit of the base 12x number.
- Seventh column ‘pattern 1’, special factor that the product of the fifth and sixth columns (the whole numbers, not just ending numbers is raised by.
- Eight column ‘pattern 2’, ending numbers of the primes and non-primes.
- Ninth column ‘pattern 3’, repeats the gap pattern in column four.
- Tenth column shows the gap between the prime being considered and the previous prime number.
- Eleventh column shows the gap between the non-prime number and previous non-prime number.
- Twelfth column shows a combination of column ten and eleven gaps.
- Thirteenth column shows the number of prime numbers lower than the non-prime number being examined.
- Fourteenth column shows the number of non-prime numbers lower than the non-prime number being examined.
- Fifteenth column shows the difference between the primes and non-prime numbers at the point of examination.
This Table shows Setaf Data from 1 to 251: –
| 1st | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| 1 | 0 | 00001 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | ||||||
| 7 | 0 | 00007 | 6 | 0 | 0 | 7 | 7 | 6 | 2 | ||||||
| 11 | 0 | 00011 | 4 | 0 | 0 | 11 | 1 | 4 | 3 | ||||||
| 13 | 0 | 00013 | 2 | 1 | 2 | 1 | 3 | 2 | 4 | ||||||
| 17 | 0 | 00017 | 4 | 1 | 2 | 5 | 7 | 4 | 5 | ||||||
| 19 | 0 | 00019 | 2 | 1 | 2 | 7 | 9 | 2 | 6 | ||||||
| 23 | 0 | 00023 | 4 | 1 | 2 | 11 | 3 | 4 | 7 | ||||||
| 29 | 0 | 00029 | 6 | 2 | 4 | 5 | 9 | 6 | 8 | ||||||
| 31 | 0 | 00031 | 2 | 2 | 4 | 7 | 1 | 2 | 9 | ||||||
| 37 | 0 | 00037 | 6 | 3 | 6 | 1 | 7 | 6 | 10 | ||||||
| 41 | 0 | 00041 | 4 | 3 | 6 | 5 | 1 | 4 | 11 | ||||||
| 43 | 0 | 00043 | 2 | 3 | 6 | 7 | 3 | 2 | 12 | ||||||
| 47 | 0 | 00047 | 4 | 3 | 6 | 11 | 7 | 4 | 49 | 49 | 13 | ||||
| 0 | 49 | 00049 | 2 | 4 | 8 | 1 | 9 | 2 | 6 | 6 | 0 | -13 | |||
| 53 | 0 | 00053 | 4 | 4 | 8 | 5 | 3 | 4 | 14 | ||||||
| 59 | 0 | 00059 | 6 | 4 | 8 | 11 | 9 | 6 | 15 | ||||||
| 61 | 0 | 00061 | 2 | 5 | 0 | 1 | 1 | 2 | 16 | ||||||
| 67 | 0 | 00067 | 6 | 5 | 0 | 7 | 7 | 6 | 17 | ||||||
| 71 | 0 | 00071 | 4 | 5 | 0 | 11 | 1 | 4 | 18 | ||||||
| 73 | 0 | 00073 | 2 | 6 | 2 | 1 | 3 | 2 | 28 | 28 | 19 | ||||
| 0 | 77 | 00077 | 4 | 6 | 2 | 5 | 7 | 4 | 6 | 6 | 1 | -18 | |||
| 79 | 0 | 00079 | 2 | 6 | 2 | 7 | 9 | 2 | 20 | ||||||
| 83 | 0 | 00083 | 4 | 6 | 2 | 11 | 3 | 4 | 21 | ||||||
| 89 | 0 | 00089 | 6 | 7 | 4 | 5 | 9 | 6 | 14 | 14 | 22 | ||||
| 0 | 91 | 00091 | 2 | 7 | 4 | 7 | 1 | 2 | 8 | 8 | 2 | -20 | |||
| 97 | 0 | 00097 | 6 | 8 | 6 | 1 | 7 | 6 | 23 | ||||||
| 101 | 0 | 00101 | 4 | 8 | 6 | 5 | 1 | 4 | 24 | ||||||
| 103 | 0 | 00103 | 2 | 8 | 6 | 7 | 3 | 2 | 25 | ||||||
| 107 | 0 | 00107 | 4 | 8 | 6 | 11 | 7 | 4 | 26 | ||||||
| 109 | 0 | 00109 | 2 | 9 | 8 | 1 | 9 | 2 | 27 | ||||||
| 113 | 0 | 00113 | 4 | 9 | 8 | 5 | 3 | 4 | 28 | 28 | 28 | ||||
| 0 | 119 | 00119 | 6 | 9 | 8 | 11 | 9 | 6 | 3 | -25 | |||||
| 0 | 121 | 00121 | 2 | 0 | 0 | 1 | 1 | 2 | 14 | 14 | 4 | ||||
| 127 | 0 | 00127 | 6 | 0 | 0 | 7 | 7 | 6 | 29 | ||||||
| 131 | 0 | 00131 | 4 | 0 | 0 | 11 | 1 | 4 | 12 | 12 | 30 | ||||
| 0 | 133 | 00133 | 2 | 1 | 2 | 1 | 3 | 2 | 6 | 6 | 5 | -25 | |||
| 137 | 0 | 00137 | 4 | 1 | 2 | 5 | 7 | 4 | 31 | ||||||
| 139 | 0 | 00139 | 2 | 1 | 2 | 7 | 9 | 2 | 10 | 10 | 32 | ||||
| 0 | 143 | 00143 | 4 | 1 | 2 | 11 | 3 | 4 | 10 | 10 | 6 | -26 | |||
| 149 | 0 | 00149 | 6 | 2 | 4 | 5 | 9 | 6 | 33 | ||||||
| 151 | 0 | 00151 | 2 | 2 | 4 | 7 | 1 | 2 | 34 | ||||||
| 157 | 0 | 00157 | 6 | 3 | 6 | 1 | 7 | 6 | 18 | 18 | 35 | ||||
| 0 | 161 | 00161 | 4 | 3 | 6 | 5 | 1 | 4 | 6 | 6 | 7 | -28 | |||
| 163 | 0 | 00163 | 2 | 3 | 6 | 7 | 3 | 2 | 36 | ||||||
| 167 | 0 | 00167 | 4 | 3 | 6 | 11 | 7 | 4 | 8 | 8 | 37 | ||||
| 0 | 169 | 00169 | 2 | 4 | 8 | 1 | 9 | 2 | 6 | 6 | 8 | -29 | |||
| 173 | 0 | 00173 | 4 | 4 | 8 | 5 | 3 | 4 | 38 | ||||||
| 179 | 0 | 00179 | 6 | 4 | 8 | 11 | 9 | 6 | 39 | ||||||
| 181 | 0 | 00181 | 2 | 5 | 0 | 1 | 1 | 2 | 18 | 18 | 40 | ||||
| 0 | 187 | 00187 | 6 | 5 | 0 | 7 | 7 | 6 | 10 | 10 | 9 | -31 | |||
| 191 | 0 | 00191 | 4 | 5 | 0 | 11 | 1 | 4 | 41 | ||||||
| 193 | 0 | 00193 | 2 | 6 | 2 | 1 | 3 | 2 | 42 | ||||||
| 197 | 0 | 00197 | 4 | 6 | 2 | 5 | 7 | 4 | 43 | ||||||
| 199 | 0 | 00199 | 2 | 6 | 2 | 7 | 9 | 2 | 16 | 16 | 44 | ||||
| 0 | 203 | 00203 | 4 | 6 | 2 | 11 | 3 | 4 | 10 | -34 | |||||
| 0 | 209 | 00209 | 6 | 7 | 4 | 5 | 9 | 6 | 12 | 12 | 11 | ||||
| 211 | 0 | 00211 | 2 | 7 | 4 | 7 | 1 | 2 | 8 | 8 | 45 | ||||
| 0 | 217 | 00217 | 6 | 8 | 6 | 1 | 7 | 6 | 12 | -33 | |||||
| 0 | 221 | 00221 | 4 | 8 | 6 | 5 | 1 | 4 | 12 | 12 | 13 | ||||
| 223 | 0 | 00223 | 2 | 8 | 6 | 7 | 3 | 2 | 46 | ||||||
| 227 | 0 | 00227 | 4 | 8 | 6 | 11 | 7 | 4 | 47 | ||||||
| 229 | 0 | 00229 | 2 | 9 | 8 | 1 | 9 | 2 | 48 | ||||||
| 233 | 0 | 00233 | 4 | 9 | 8 | 5 | 3 | 4 | 49 | ||||||
| 239 | 0 | 00239 | 6 | 9 | 8 | 11 | 9 | 6 | 50 | ||||||
| 241 | 0 | 00241 | 2 | 0 | 0 | 1 | 1 | 2 | 26 | 26 | 51 | ||||
| 0 | 247 | 00247 | 6 | 0 | 0 | 7 | 7 | 6 | 10 | 10 | 14 | -37 | |||
| 251 | 0 | 00251 | 4 | 0 | 0 | 11 | 1 | 4 | 6 | 6 | 52 |
The Initial Difference
The difference starts at -13, meaning that there are thirteen primes lower than the first non-prime 49 (note that prime 5 is not included nor 2, or 3, as previously explained).
Sequence 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, consists of these first 13 prime numbers in my selective number line.
They appear in the difference column as a negative value -13.
This negative difference continues to increase in negative size as we go down the list, which the table shows as -37. 37 is the number of primes less than the odd-composite non-prime of 247. To see further along the number line than this point, please examine the spreadsheet.
Excerpt from the beginning of the ‘Main’ spreadsheet showing the Difference values (i.e. the Data) as shown in the table above: –
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: 32.55 Mb and 122.67 Mb.
You may download the spreadsheet file from this address.
- Address – https://www.mediafire.com/folder/ztev7t9nh0xk3/Spreadsheet+for+PNonP+Patterns.
- The spreadsheet has “Main” in its title /filename.
Should you find that the spreadsheet downloads as a “read only file”, usually you can simply rename it on your own computer, then open it to use it in an “editable” form.
Difference between primes and non-primes at the point of examination
When the difference is -78 between non-primes and primes, there are 178 non-primes and 256 primes lower than this point of examination. At this point on row 444; the list is showing non-prime 1631.
This is the lowest negative difference. The difference now starts to “rise”; i.e. decrease towards the positive.
The Most Negative Difference
- (Please note that row numbers may change slightly according to the configuration of the spreadsheet).
The Prime to Non-Prime Difference Balances Out at Zero
At row 1360 and non-prime 4891 the difference balances out as zero. This is the (first) examination point where the number of primes and non-primes match at zero. There are 652 non-primes and 652 primes below this certain number. After this row, primes and non-primes increase at rates so that the difference between them becomes a positive number.
The non-primes build to +1. However the difference immediately dips back to zero for a number of rows. It then progresses in the positive again.
If this information was represented on a graph it would be seen that the detail of the progression would be anything but smooth. The values waver up and down, but by comparatively small amounts as shown below.
Once past the equalisation point of zero difference, the non-primes are always increasing in a greater amount relative to the primes. The difference value gradually becomes positively greater.
This positive difference means that there are greater and greater number of non-primes compared with the prime numbers the further up the number line we go.
End of the Spreadsheet
At row 70004, non-prime 262481 is the farthest point I have taken the checking of the prime numbers right now. The nearest prime to this non-prime is 262459.
There are now 22969 primes and 47019 non-primes lower than this point. The difference is 24050 between prime and non-prime.
Variation in the Difference Value
There are in fact several variation points where one might suppose that the difference is decreasing or increasing, only to see a few rows later that the opposite to expectations has happened for a few rows.
The Ratio Spreadsheet
In order to take this further I have decided to develop a separate spreadsheet to look at the differences and the ratio between them. This spreadsheet will have “Ratio” in its title /filename.
