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A new paradigm for studying prime numbers
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18-06-2019 12:54am#1Here's a promising new paradigm for the study of prime numbers.
The general theory is that prime numbers probably continue occurring at slightly lower frequencies all the way to the realms of huge integers, as established by computer-assisted frequency counts showing that even when you reach integers as large as one billion, the frequency of primes in blocks of 1,000 (or any arbitrary value) remains well above 5% -- a good discussion appears here:
http://phillipmfeldman.org/mathematics/primes.html
and this statistic decreases towards what many suspect may be an asymptote, although it could just decrease so slowly that infinity never practically overwhelms it (a situation where primes would become so infrequent at large integer ranges that finding even one might become too laborious for even a super-computer).
If there were to be an asymptote involved it could be a fairly small value too, but it will take perhaps decades of supercomputer expansion to hint at its value.
These are just contextual observations that have no direct bearing on the paradigm I am going to introduce here.
This is the "blocks of 30" paradigm which equates to blocks of 15 consecutive odd numbers (the first one being 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29).
Because 30 is 3x5x2, some of these values will never contain prime numbers; in the first block, 3 and 5 are primes while 9, 15, 21, 25 and 27 are non-primes. In every subsequent block, 3+30n and 5+30n will also be non-primes, as will the other numbers mentioned (9+30n, 15+30n, 21+30n, 25+30n and 27+30n). These are what I call the seven excluded columns, leaving eight columns where prime numbers (higher than 5) could appear.
In the first block, 1 is defined to be non-prime and 2 is the lone even non-conforming example of a prime number.
But our interest would be in studying whether there was any predictable qualities involved in the occasional exclusion of primes in the eight columns that are friendly territory for primes.
The first excluded value is 49 in the 19 column (block 2). The second excluded value is 77 in the 17 column (block 3). The third excluded value is 91 in the 1 column (block 4). The fourth is 119 (in the 29 column of block 4) and the fifth is 121 (in the 1 column of block five).
These excluded non-primes occur when smaller primes (but larger than 5) multiplied by either themselves or by higher prime(s) hit a value in the "friendly columns." The first five examples are 7x7, 7x11, 7x13, 7x17 and 11x11. The next five would be 7x19, 11x13, 7x23, 13x13 and 11x17.
The questions of interest would be:
(a) do the excluded values appear at random in the eight friendly columns, or do they populate some columns more frequently than others?
(b) is there any friendly column that is never visited (at least in the testable range) by excluded values? (this is answered below as "no" as established within the first ten excluded values).
(c) are there any rules for predicting where the excluded values (non-primes) will show up?
(d) How frequently do excluded values occur in successive blocks? The existence of the friendly columns establishes that there can never be more than 8 primes in every 30 integers or 15 odd numbers, past 31. If there is always at least one excluded number (like 49 in block 2) then there can never be more than 7/15 primes among blocks of odd numbers that start at 30n+1. But what rules are suggested for frequencies of primes as the hits start increasing? Do they increase at any appreciable rate? (the rate is obviously not going to increase rapidly since generating primes are spaced further apart by exclusion which would space their products further apart, albeit from increasing numbers of generators).
Below, I test out where the first 20 excluded values appear. Note that I have left out block one with its non-conforming values of 2, 3 and 5. I also test out the number of excluded values per block.
Non-primes in otherwise friendly columns for primes
BLOCK __ 1 ___ 7 ___ 11 ___13___17___19___23___29 ____ n excluded values
__ 2 ____ 31 __ 37 __ 41 __ 43 __ 47 __ 49 __ 53 __ 59 ____ 1
__ 3 ____ 61 __ 67 __ 71 __ 73 __ 77 __ 79 __ 83 __ 89 ____ 1
__ 4 ____ 91 __ 97__101__103__107__109 __113__119 ____ 2
__ 5 ____121__127__131__133__137__139__143__149 ____ 3
__ 6 ____151__157__161__163__167__169__173__179 ____ 2
__ 7 ____181__187__191__193__197__199__203__209 ____ 3
__ 8 ____211__217__221__223__227__229__233__239 ____ 2
__ 9 ____241__247__251__253__257__259__263__269 ____ 3
__10 ___ 271__277__281__283__287__289__293__299 ____ 3
________________________________________
The excluded numbers above are replaced here by their prime factors to show that all cases have been identified (any unidentified would have to be products of other primes and it can be seen that no such cases occur).
BLOCK __ 1 ___ 7 ___ 11 ___13___17___19___23___29 ____ n excluded values
__ 2 ____ 31 __ 37 __ 41 __ 43 __ 47 __7*7 __ 53 __ 59 ____ 1
__ 3 ____ 61 __ 67 __ 71 __ 73 __7*11__ 79 __ 83 __ 89 ____ 1
__ 4 ____7*13__ 97__101__103__107__109 __113__7*17____ 2
__ 5 ____11*11_127__131__7*19_137__139_11*13_149 ____ 3
__ 6 ____151__157__7*23 _163__167__13*13_173__179 ____ 2
__ 7 ____181__11*17_191__193__197__199__7*29_11*19____3
__ 8 ____211__7*31_13*17_223__227__229__233__239 ____ 2
__ 9____241__13*19_251__11*23_257_7*37 _263__269 ____ 3
__10 ___271__277__281__283__7*41_17*17_293__13*23 ___ 3
______________________________________________________
The frequency of excluded numbers will be dominated by multiples of 7 which are going to appear roughly three times in each two blocks until the spacing of prime numbers increases. Each new set will be spaced by at least the value of the newly introduced "excluder" -- for example, when 11 appears, it could then appear again as early as five columns later but will miss some occasions in the already excluded columns (e.g. 11x15 between 11x13 and 11x17).
Each new excluder will generate new sets of excluded numbers. There is no chance that an excluded number could have two different sets of prime factors, although the proof of that might require some work.
If we assume that each excluder will generate new excluded non-primes for half of its odd multiples, then the frequency of excluded numbers must keep increasing at a fairly steady pace, until it reaches equilibrium when the spacing of primes cancels out the new excluded numbers.
I will leave it at that for now and see if any further logic can be derived for this paradigm. There is no clear indication from the small sample offered that there might be any order to the appearance in each column of occasionally excluded non-primes. But already it looks as though the frequency is fairly well distributed in the eight prime-friendly columns.1
Comments
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To give a larger sample of non-primes in these eight columns, I have added blocks 11 to 20 below ... the non-primes are shown by factors rather than their actual values, which can be derived from the adjacent numbers in columns and rows:
BLOCK ___ 01 __ 07 __ 11 __ 13 __ 17 __ 19 __ 23 __ 29 ___ n excluded
__ 11 ____7*43_307 _ 311 _ 313 _ 317_11*29_17*19_7*47____ 4
__ 12 ____331 _ 337 _11*31_343 _ 347 _ 349 _ 353 _ 359 ____ 1
__ 13 ___19*19_367 _7*53_ 373 _13*29_379 _ 383 _ 389 ____ 3
__ 14 ___17*23_397 _ 401_13*31_11*37_409 _7*59_ 419 ____ 4
__ 15 ____421 _7*61_ 431 _ 433 _19*23_ 439 _ 443 _ 449 ____ 2
__ 16 ___11*41_457 _ 461 _ 463 _ 467 _7*67_11*43_479 ____ 3
__ 17 ___13*37_487 _ 491 _17*29_7*71_ 499 _ 503 _ 509 ____ 3
__ 18 ___7*73_11*47_ 521 _523_17*31_23*23_13*41_11*49___ 6
__ 19 ____541 _ 547_19*29_7*79_ 557 _13*43_ 563 _ 569 ____ 3
__ 20 ____571 _ 577 _7*83_11*53_ 587 _ 589 _ 593_19*31____ 3
_____________________________________________________
This is the frequency count of excluded non-primes in each of the eight friendly columns through 19 blocks (2-20) ...
Column ___ 01 _ 07 _ 11 _ 13 _ 17 _ 19 _ 23 _ 29
Frequency __ 8 __ 5 __ 6 __ 6 __ 7 __ 8 __ 6 __ 6
(if you counted 1 as an excluded non-prime, the frequency would be 9 for 01).
This is well within the variability one might expect of a set of numbers converging on equal frequency in a large sample going forward.
This gives 52 excluded nonprimes in the 285 odd numbers 31 to 599. There are also 133 automatically excluded in the seven hostile columns, so the net result is 185 excluded out of 285, or 100 primes 31 to 593 inclusive.
The frequency of excluded non-primes averages 1 for blocks 2 and 3, and one could argue that it averages 1 for blocks 1 to 3.
The frequency from block 4 to block 10 is 18 for an average of 2.57 in those seven blocks.
The frequency from block 11 to 15 is 14 for an average of 2.8 per block, then from 16 to 20 it jumps to 18, an average of 3.6 per block.
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Is this published anywhere? Original research?0
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Not published as far as I know, would not claim it to be that big of a breakthrough in our understanding of prime numbers, just a possible pathway to greater understanding if any patterns could be found. Probably just a few more unsolvable "probably true" theorems about prime numbers to add to the list.
My guess is that the "friendly columns" would have very similar numbers of primes and nonprimes eventually, let's say after about 10,000 groups (to 299,999). Whether there would be any reason for them to be predictable is another thing, they might just converge on similar values without ever being exactly the same. From the percentages shown in the linked study, I would say that eventually the groups of eight friendly columns would be full of excluded nonprimes on many occasions and would allow in one or two primes every so often, but apparently you still get the clusters of three or even four allowed values in a row as primes.
There is a "formula" that defines primes and non-primes in a sequential way but it does not transform into an equation that can be solved.
The formula is that from 3 up, the odd numbers are non prime every time the relation an+2a is verified (where a and n are odd numbers). For example, 3n+6 will always be non-prime. 5n+10, 7n+14, 9n+18, 11n+22 etc, always non prime. If any odd number that verifies an+2a is given the value A and every other odd number the value 1, then the number that defines prime or non prime odd numbers (starting from 3) would be derived by adding the excluder for multiples of 3 to those for multiples of 5, 7, 9, 11 etc with the logic being that addition would be defined as allowing any value to be only 1 or A, for it to be 1 then all terms would have to be 1, for it to be A, any term could be A.
so for the first 60 odd numbers starting at 3 and ending at 121, the terms being added would be
111A11A11A11A11A11A11A11A11A11A11A11A11A11A11A11A11A11A11A11
011111A1111A1111A1111A1111A1111A1111A1111A1111A1111A1111A111
001111111A111111A111111A111111A111111A111111A111111A111111A1
0001 etc
(makes no difference if you include or exclude terms generated from non-prime starters like 9 as they will just have their A values in same spots as an earlier more densely populated term, the first one 3 in this case).
then for 11 ...
000011111111111A1111111111A1111111111A1111111111A1111111111A
You could add further terms but as no odd number smaller than 121 has factors that are both larger than 11, these terms will suffice.
Visually you can't easily see how they add but looking at the last value representing 121, it's 1+1+1+1+A which in the logic used is A.
For 13 which is the sixth value in each row, the sum is 1+1+1+1+1 which is 1 (prime number).
The entire set of odd numbers from 3 up to infinity would be defined as prime or non prime by adding such terms ... a segment well into the set at a number which is 311 x 313 (both primes) would look something like this (showing only the terms from that number on):
terms smaller than 311 __ none would hit A at this value
term 311 __ A (followed by 310 1's) A
term 313 __ A (followed by 312 1's) A
terms larger than 313 up to the value itself __ none would hit A at this value.
I don't know if a computer could be programmed to add these terms or if this is already known and being done.
The logic would be something like "inspect each column, and if any A, then sum is A, if all terms zero or one (zero is used as a placeholder to start the term as shown in my examples for 121) ... so if all terms zero or one, the sum is one (meaning prime number)."
This procedure would also identify factors, although with the glitch that factors containing smaller factors would not be fully operational. For example, the procedure would show that 45 had factors 3, 5 and 9 but the number is a product of 5 and 9.0 -
A more complex paradigm would remove all multiples of seven. To do that, you have to consider groups of 105 odd numbers (or 210 integers) and so the "prime friendly columns" would be all those from the first seven groups in the first paradigm, except those that were multiples of 7.
So the friendly columns are reduced from
1 7 11 13 17 19 23 29 31 37 41 43 47 49 53 57 etc to
1 11 13 17 19 23 29 31 37 41 43 47 53 57 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
which removes 7, 49, 77, 91, 119, 133, 161 and 203 which now become hostile columns as we repeat every set of 105.
That is a total of 48 friendly columns and 57 hostile to primes (7 now has the same first-time exception as 3 and 5).
It would take a lot more work to establish the frequency of nonprimes in these 48 friendly columns.
The only nonprimes in the first block would be 1, 121, 143, 169, 187 and 209 (6 excluded)
Those in the second block would be 11, 37, 43, 79, 89, 109, 113, 131, 151, 167, 181, 193 and 197. (13 excluded)
In the third block would be 17, 31, 53, 61, 73, 97, 107, 109, 113, 131, 139, 163, 179, 191, 199 (15 excluded)
Of those only 109, 113, 131 have two excluded, while 13, 19, 23, 29, 41, 47, 59, 67, 71, 83, 101, 103, 127, 137, 149, 157, and 173 have yet to exclude a nonprime (to 629).0 -
I checked the distribution of non-primes in the 48 "prime friendly" columns in the 105-odd-number sets, as described above, to the end of row eleven (2101 to 2309) and found that all 48 columns received at least one non-prime. The least populated columns were 41 (one nonprime at 671, 61x11) and 199 (2299, 209x11).
So it was at 2299 that the system established that no friendly column would remain all prime numbers. Most columns by that time had three to six excluded nonprimes.
Out of 48 possible primes, the first three blocks (rows) as already mentioned excluded 6, 13 and 15 nonprimes. From there on, the numbers excluded were 16, 18, 19, 21, 17, 21, 21, 21.
The frequency of nonprimes per 48 friendly columns is therefore approaching half by 2299.
Checking a table of primes to 10,000, I inspected the 41st row of this set, which is 8401 to 8609. There were 21 primes and therefore 27 nonprimes by this level. The frequency of nonprimes is therefore increasing quite slowly.
By row 4760 which runs from 999,601 to 999,809, there were 15 primes and 33 excluded nonprimes.0 -
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Thinking that if there were any useful patterns to be uncovered, they might show up in the first fifty sets as described above, this table shows just the prime numbers that occupy rows 12 to 48 taking us to all primes below 10,000. Set 12 is 2311 to 2519. Primes are shown as column headers, nonprimes are shown as xx. (example, 001 in set 12 means 2311 is prime, next four are nonprime, 2333, 2339, 2341, 2347 and 2351 are all primes, etc. )
SET __01 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97101 03 07 09 13 21 27 31 37 39 43 49 51 57 63 67 69 73 79 81 87 91 93 97 99 209 __ n primes (/48)
12 ___ 01 xx xx xx xx 23 29 31 37 41 xx 47 xx xx 61 67 71 73 79 83 89 xx101 xx 07 xx 13 xx 27 31 37 xx xx 49 xx 57 63 67 xx xx xx xx xx xx 93 xx xx xxx ___ 25
13 ___ 01 11 xx xx 19 23 29 31 37 xx xx xx xx 59 xx xx 71 73 xx xx 89 97101 xx xx xx 13 xx 27 xx 37 39 43 xx 51 57 63 67 69 73 79 xx 87 91 93 xx 99 209 ___ 30
14 ___ 01 11 xx xx 19 23 xx xx 37 xx xx 47 xx 59 61 67 71 73 xx xx 89 xx xx 03 07 xx 13 21 27 31 xx xx xx 49 xx 57 xx 67 xx 73 79 xx 87 xx xx 97 xx 209 ___ 26
15 ___ xx xx 13 17 xx 23 29 31 xx xx xx xx xx 59 61 xx 71 xx 79 83 xx 97101 xx xx 09 xx 21 27 xx xx 39 43 49 xx xx xx xx 69 xx 79 81 xx xx xx 97 xx xxx ___ 22
16 ___ xx xx 13 17 19 xx xx 31 37 41 xx xx 53 59 xx 67 71 xx 79 xx xx xx101 03 07 09 xx 21 xx xx xx xx xx 49 51 57 63 xx 69 73 79 81 xx xx 93 97 xx 209 ___ 27
17 ___ 01 11 13 xx xx xx 29 31 xx xx xx 47 53 xx xx xx xx 73 xx xx 89 97101 03 07 09 xx xx xx 31 xx 39 xx xx 51 57 xx 67 69 73 79 81 87 xx xx 97 99 xxx ___ 26
18 ___ 01 11 13 xx xx 23 xx xx 37 xx 43 47 53 xx 61 67 xx 73 xx xx 89 xx101 03 07 xx xx 21 27 31 xx 39 xx 49 xx 57 63 xx 69 xx xx xx xx 91 xx 97 99 209 ___ 27
19 ___ xx xx 13 17 xx 23 xx xx xx 41 43 xx 53 xx xx 67 71 73 xx 83 xx 97101 xx xx 09 xx xx 27 31 37 39 43 49 51 xx 63 67 xx xx xx xx 87 xx xx xx xx 209 ___ 24
20 ___ 01 11 13 xx xx 23 29 31 37 xx xx xx xx 59 61 67 xx xx xx 83 89 xx101 03 xx 09 xx 21 xx xx 37 39 43 49 xx xx 63 67 69 xx xx xx 87 xx xx xx xx xxx ___ 24
21 ___ xx 11 xx 17 19 xx 29 31 xx 41 43 xx 53 59 61 xx 71 73 xx 83 89 97xxx xx xx xx xx xx 27 xx 37 39 xx 49 xx 57 63 xx xx 73 xx xx xx 91 xx 97 xx 209 ___ 25
22 ___ xx 11 13 xx xx xx xx 31 37 41 xx 47 53 xx xx xx 71 73 xx 83 xx 97xxx 03 07 09 13 xx xx xx 37 39 xx xx 51 57 xx xx xx 73 xx 81 87 xx 93 xx xx xxx ___ 23
23 ___ 01 xx xx 17 19 23 29 31 37 xx 43 xx 53 59 xx xx 71 xx xx 83 xx xx101 03 xx 09 13 xx xx 31 xx 39 xx xx xx xx 63 67 69 73 79 81 xx xx 93 97 xx xxx ___ 26
24 ___ 01 xx xx xx xx xx xx 31 xx 41 xx 47 xx 59 xx xx xx 73 79 xx 89 xx101 03 07 xx 13 21 27 xx 37 39 43 xx xx 57 63 xx 69 73 79 81 xx 91 93 xx xx 209 ___ 26
25 ___ xx 11 xx xx 19 xx xx xx 37 41 xx 47 xx 59 61 67 xx 73 79 xx xx xx xx xx 07 xx 13 xx 27 31 xx 39 xx 49 xx 57 xx xx 69 xx xx xx 87 91 93 97 xx xxx ___ 22
26 ___ xx 11 xx xx 19 xx xx xx 37 41 xx 47 xx 59 61 67 xx 73 79 xx xx xx xx xx 07 xx 13 xx 27 31 xx 39 xx 49 xx 57 xx xx 69 xx xx xx 87 91 93 97 xx xxx ___ 22
27 ___ xx 11 xx 17 19 23 xx xx xx 41 43 47 xx 59 61 67 71 xx xx xx xx 97xxx 03 xx 09 13 21 xx 31 xx xx xx xx xx xx 63 xx xx xx 79 81 87 91 93 97 99 209 ___ 26
28 ___ xx xx 13 xx 19 23 xx 31 xx 41 xx 47 xx xx xx 67 71 73 79 xx xx xx xx xx xx 09 13 21 xx 31 37 xx 43 xx 51 57 xx xx 69 73 79 81 87 91 xx 97 99 209 ___ 27
29 ___ 01 xx xx 17 xx 23 xx xx xx xx 43 47 xx 59 xx xx xx 73 xx xx xx xx101 xx 07 xx xx xx 27 31 xx xx xx 49 xx 57 63 67 xx 73 xx xx 87 xx 93 xx 99 209 ___ 20
30 ___ 01 11 xx xx xx 23 xx 31 xx 41 43 xx 53 xx 61 xx xx 73 xx 83 xx xx xx xx 07 09 13 21 27 31 xx 39 xx xx xx 57 xx 67 xx 73 79 81 87 xx xx 97 xx 209 ___ 25
31 ___ 01 11 xx 17 xx 23 29 xx 37 xx 43 xx 53 59 61 67 xx 73 79 xx 89 97xxx xx xx xx xx 21 27 xx xx xx xx 49 51 xx xx xx 69 73 xx 81 xx 91 xx xx xx xxx ___ 23
32 ___ xx 11 xx xx 19 xx xx xx 37 41 43 xx 53 59 61 67 71 xx xx xx 89 97xxx xx xx 09 xx xx 27 xx xx xx 43 49 51 xx 63 xx 69 xx 79 81 xx 91 93 xx 99 209 ___ 25
33 ___ xx xx 13 17 xx xx xx xx xx 41 43 xx xx 59 61 xx 71 73 xx 83 xx xx xx 03 07 09 13 21 xx xx 37 xx 43 49 51 xx 63 xx xx xx 79 xx 87 91 xx 97 xx xxx ___ 23
34 ___ xx xx xx 17 19 xx 29 31 37 41 xx 47 53 xx 61 67 71 xx xx 83 89 97xxx xx xx 09 13 xx 27 xx xx 39 xx 49 xx xx xx xx xx 73 79 xx xx 91 xx 97 99 xxx ___ 24
35 ___ xx 11 xx xx 19 xx xx xx 37 xx xx 47 53 xx xx 67 71 73 79 xx 89 97xxx 03 07 xx 13 xx xx xx xx xx 43 xx xx 57 xx 67 69 xx xx 81 xx 91 93 xx xx 209 ___ 22
36 ___ 01 xx xx xx 19 xx xx xx xx xx 43 xx xx xx 61 67 xx xx xx 83 xx xx101 xx 07 09 xx xx 27 31 37 39 xx 49 xx 57 xx 67 xx 73 79 xx 87 91 xx 97 99 209 ___ 23
37 ___ 01 xx 13 17 xx 23 29 31 xx xx 43 47 xx xx 61 xx xx xx 79 83 89 xx xx xx xx 09 13 21 27 31 xx 39 43 xx xx 57 63 67 xx xx xx 81 xx xx 93 97 99 xxx ___ 26
38 ___ xx xx xx xx 19 23 xx xx xx xx xx 47 53 59 xx xx 71 xx xx 83 xx 97xxx 03 07 09 13 xx xx 31 37 xx xx 49 xx 57 63 67 xx xx 79 81 xx xx 93 xx xx xxx ___ 21
39 ___ xx xx 13 xx xx xx 29 31 37 xx xx xx xx 59 xx xx xx 73 79 xx 89 xx101 xx 07 09 13 21 xx 31 37 xx 43 xx xx xx xx 67 xx xx xx 81 87 91 xx xx 99 xxx ___ 21
40 ___ 01 xx xx xx 19 xx 29 31 xx 41 43 47 53 xx xx xx xx 73 79 83 xx 97101 03 07 xx xx 21 27 xx xx 39 xx xx xx xx 63 xx xx 73 79 xx 87 xx xx 97 99 xxx ___ 24
41 ___ xx xx xx xx 19 23 29 31 xx xx 43 47 xx xx 61 67 xx xx xx xx xx xx101 xx xx xx 13 21 27 xx 37 39 43 xx xx xx 63 xx xx 73 xx 81 xx xx xx 97 99 209 ___ 21
42 ___ xx xx 13 17 19 xx xx 31 37 xx xx xx 53 59 xx 67 71 xx 79 83 89 97xxx 03 xx 09 xx 21 27 31 37 xx 43 xx 51 xx xx xx 69 73 xx xx xx xx 93 97 xx 209 ___ 26
43 ___ 01 11 xx 17 19 xx 29 xx xx 41 43 47 xx xx xx 67 xx 73 xx xx xx xx xx 03 xx 09 13 21 xx 31 xx xx 43 49 51 xx xx xx xx xx 79 81 87 91 93 xx xx 209 ___ 24
44 ___ xx 11 13 xx 19 xx 29 xx 37 xx xx xx xx xx 61 xx xx 73 79 xx xx 97xxx 03 07 xx xx 21 27 31 xx xx 43 xx 51 57 xx xx 69 73 79 xx xx 91 xx 97 xx 209 ___ 23
45 ___ 01 xx xx 17 xx xx xx xx 37 41 43 xx 53 xx xx xx 71 xx 79 83 xx 97101 03 xx 09 xx xx xx 31 37 xx xx xx 51 57 63 xx xx 73 79 81 xx 91 93 97 99 xxx ___ 25
46 ___ xx 11 13 17 xx 23 29 xx xx 41 xx 47 xx xx 61 xx 71 xx xx 83 89 97101 xx xx xx xx xx xx xx 37 xx xx xx 51 xx 63 xx 69 73 79 81 xx xx 93 xx 99 xxx ___ 22
47 ___ 01 xx xx 17 19 xx 29 xx 37 xx xx xx xx 59 61 xx xx 73 79 83 89 xx xx xx 07 09 xx 21 27 31 xx xx 43 xx 51 57 xx xx 69 73 79 xx xx 91 xx 97 99 xxx ___ 25
48 ___ 01 xx 13 17 xx xx xx 31 37 xx xx xx 53 59 61 xx 71 xx 79 xx xx 97xxx 03 xx xx xx xx xx xx 37 39 xx xx xx xx xx 67 69 xx xx xx xx 91 xx 97 99 209 ___ 20
This takes us up to 10,079. The friendly columns for primes are collecting non-primes at fairly equal rates, I am going to study this output and see if any patterns can be found.0 -
The post on 20th of June has an error in it, 57 as a prime should have read 59.
That error fortunately does not appear in the actual analysis that follows where 59 is correctly identified.
One principle that emerges is symmetry -- each friendly column has its partner on the other side of 105, e.g. 103 and 107, 101 and 109, all 24 pairs add up to 210.
1+209 ... 11+199 ... 13+197 ... 17 +193 ... 19 +191 ... 23 + 187 ... 29 + 181 ... 31 + 179 ... 37 + 173 ... 41 +169 ... 43 + 167
47 + 163 ... 53 +157 ... 59 + 151 ... 61 +149 ... 67 + 143 ... 71 + 139 ... 73 + 137 ... 79 + 131 ... 83 + 127 ... 89 + 121 ...
97 + 113 ... 101 + 109 ... 103 + 107
=====================================
In other developments, I looked up tables of prime numbers of about 1,000 billlion, and found that the spacing of primes up that high was not a lot further apart than at 1 million. The average set of 48 friendly columns up that high will yield about 40 or 41 nonprimes and 7 or 8 primes. It makes me wonder if the spacing ever reaches equilibrium so that the frequency of primes past a certain point either flattens out or falls so slowly as to be nearly imperceptible. The reason why it might do so is that new primes entering into the task of providing multiples are so few and far between that their nonprime products will be avoiding more and more possible landing spots. But I then realized that the frequency has to keep dropping at least slightly.
Take the case of 11, the first prime that can produce hits in this system (3, 5 and 7 are excluded from play so to speak).
Every 11th set will have its 11-excluded values in the same spot. Compare row 12 above with row 23 ... the nonprimes that are multiples of 11 here are shown with XX instead of xx. They are all in the same columns.
12 ___ 01 XX xx xx xx 23 29 31 37 41 xx 47 xx xx 61 67 71 73 79 83 89 xx101 xx 07 xx 13 XX 27 31 37 xx XX 49 xx 57 63 67 xx xx xx xx XX xx 93 xx xx XXX ___ 25
23 ___ 01 XX xx 17 19 23 29 31 37 xx 43 xx 53 59 xx xx 71 xx xx 83 xx xx101 03 xx 09 13 XX xx 31 xx 39 XX xx xx xx 63 67 69 73 79 81 XX xx 93 97 xx XXX ___ 26
This extended to all other primes entering the system means that groups of rows will contain equal numbers of non primes plus those that are new (starting with prime number squared).
It is staggering how many primes participate by a given prime times all of the lower primes.
For example, just going as high as 23,
3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 = 111,546,435 which is greater than the square of 10079, the largest prime in row 48.
This means that every prime in the tables above (about 1300 in total) would be participating in the table to that value.
If we then add in 29, 31, 37, 41 and 43, the product is over 500 million billion.0
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