I have realized much of the information I have provided requires a greater level of context and simplification. This is the beginning of the effort to provide better context with a much simpler foundation. My idea or proposal that the infinite series of all natural numbers has an equality as an infinite product of prime geometric series, just as Euler's product is an infinite product of inverse prime geometric series, requires a simpler foundation and mathematical logic. That simpler foundation and mathematical logic is the 'Sieve of Eratosthenes'.
I will use the explanation for the the 'Sieve of Eratosthenes' from Wikipedia, https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes.
OverviewA prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself.To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method:
- Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n).
- Initially, let p equal 2, the smallest prime number.
- Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the list (these will be 2p, 3p, 4p, ...; the p itself should not be marked).
- Find the smallest number in the list greater than p that is not marked. If there was no such number, stop. Otherwise, let p now equal this new number (which is the next prime), and repeat from step 3.
- When the algorithm terminates, the numbers remaining not marked in the list are all the primes below n.
The main idea here is that every value given to p will be prime, because if it were composite it would be marked as a multiple of some other, smaller prime. Note that some of the numbers may be marked more than once (e.g., 15 will be marked both for 3 and 5).
For the data that we are interested in from the 'Sieve of Eratosthenes', we want the sets of numbers that are the result of each iteration, and only the numbers that have not been previously marked. We also want to know what the remainder set is after each iteration. The list of integers that we begin with also starts with the number 1 and therefore the number 1 will always be in each remainder set.
Example: For the prime number 3 iteration, the original sieve marks out 3,6,9,12,15,18,21,24,27,... However 6,12,18,24 have already been marked out by the previous prime number 2 iteration. We only want the set of numbers from each iteration that have not been previously marked out. Also, we want to include the prime number in each iteration as a member of that iteration's sieve set. The easiest method to accomplish this is to only mark out multiples of the next prime number in the previous iteration's remainder set.
Here is a table with the 'Sieve of Eratosthenes' sets we are looking for marked out.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
| 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 |
| 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 |
| 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 |
| 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 | 180 |
| 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |
| 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 |
Legend:
| Prime | Set Name | SOE Sieve Set | Remainder Name | SOE Sieve Remainder Set |
|---|---|---|---|---|
| 2 | soe2 |
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40, 42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78, 80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112, 114,116,118,120,122,124,126,128,130,132,134,136,138,140, 142,144,146,148,150,152,154,156,158,160,162,164,166,168, 170,172,174,176,178,180,182,184,186,188,190,192,194,196, 198,200,202,204,206,208,210,... |
soe2_rem |
1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39, 41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77, 79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111, 113,115,117,119,121,123,125,127,129,131,133,135,137,139, 141,143,145,147,149,151,153,155,157,159,161,163,165,167, 169,171,173,175,177,179,181,183,185,187,189,191,193,195, 197,199,201,203,205,207,209,... |
| 3 | soe3 |
3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99,105,111, 117,123,129,135,141,147,153,159,165,171,177,183,189,195, 201,207,... |
soe3_rem |
1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59, 61,65,67,71,73,77,79,83,85,89,91,95,97,101,103,107,109,113, 115,119,121,125,127,131,133,137,139,143,145,149,151,155,157, 161,163,167,169,173,175,179,181,185,187,191,193,197,199,203, 205,209,... |
| 5 | soe5 | 5,25,35,55,65,85,95,115,125,145,155,175,185,205,... | soe5_rem |
1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73, 77,79,83,89,91,97,101,103,107,109,113,119,121,127,131,133, 137,139,143,149,151,157,161,163,167,169,173,179,181,187,191, 193,197,199,203,209,... |
| 7 | soe7 | 7,49,77,91,119,133,161,203,... | soe7_rem |
1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83, 89,97,101,103,107,109,113,121,127,131,137,139,143,149,151, 157,163,167,169,173,179,181,187,191,193,197,199,209,... |
Interestingly, the soe sets and remainder sets are all periodic and symmetric.
This is easy to notice starting in the soe3_rem remainder set.
soe3_rem: {1,5} for the primorial length of 6.
{1,5} +6n, where n = 0→∞
*** This is where we get the statement that every prime number greater than 3, must be of the form 6n ±1
soe5 set: {5,25} for the primorial length of 30.
{5,25} +30n, where n = 0→∞
soe5_rem set: {1,7,11,13,17,19,23,29} for the primorial length of 30.
{1,7,11,13,17,19,23,29} +30n, where n = 0→∞
*** The soe5_rem set is the foundation of Gary Croft's: 'The Prime Spiral Sieve' https://www.primesdemystified.com
soe7 set: {7,49,77,91,119,133,161,203} for the primorial length of 210.
{7,49,77,91,119,133,161,203} +210n, where n = 0→∞
soe7_rem set: {too big to fit here, see below} for the primorial length of 210.
{1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,151,157,163,167,169,173,179,181,187,191,193,197,199,209} +210n, where n = 0→∞
*** The soe7_rem set is the foundation of a recent paper entitled “The Periodic Table of Primes" [4]. Where the authors argue that primes can be predicted using a periodic table-like structure."
These 'Sieve of Eratosthenes' sets and remainder sets are also the basis for James McCanney's work, 'New Definition of Prime Numbers with Sppn Tables and Proofs by Induction'.
These 'Sieve of Eratosthenes' sets and remainder sets are also the basis for 'Liu Fengsui's Prime Formula' and subsequent papers.
The 'Sieve of Eratosthenes' sets and remainder sets have a lot of interesting properties. However we are interested in a different property of these sets and that will be the topic of my next post.
Continuation: The 'The Sieve of Eratosthenes - Part 2
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