sw00887math--my-paired-twin-primes-proof

31


 is nothing fancy, but a bit complex.  The goal is to discern
how/where/why paired twin primes can be found.  Here is
a simple example using decade two.  (The first decade is:
0 1 2 3 4 5 6 7 8 9.)  (In a decade pattern scheme, decade
10 is actually a 1 because digit addition is used to determine
decade patterns.)

10   |   11      12      13      14      15      16      17      18      19

Note:  Only an integer whose final digit is "5" can be at the center of
a paired twin primes set because an integer whose final digit is "5"
will be encountered if the center is at an integer whose final digit is
"1" or "3" or "7" or "9" (11 + 4 = 15  13 + 2 = 15  17 - 2 = 15  19 - 4 = 15).

Step 1.  That integer whose final digit is "5" must be 
              evenly divisible by "3".
Step 2.  In the decade being investigated/ the integers
              whose final digits are "2" and "8" must be
              evenly divisible by "6".
Step 3.  Also, one of the integers noted in Step 2
              must be evenly divisible by "4".  In the
              example/ that integer is 12.  Interestingly,
              in every even decade "4" divides evenly
              into the final-digit-2 integer; and in every
              odd decade "4" divides evenly into the
              final-digit-8 integer.  [ Formula for finding
              Decade #: (decade's final-digit-5 integer
              + 5) divided by 10 ]         (285 + 5)/10

Note 1  --- This is where complexity arrives. ---
Example: 280 281 282 283 284 285 286 287 288 289
Odd-integer squares follow this final-digit pattern:  1 9 5 9 1  
as in  1 x 1 = 1, 3 x 3 = 9, 5 x 5 = 25, 7 x 7 = 49, 9 x 9 = 81.
Three ways to find them are: 
1. (a natural number summation sequence term times 8) + 1
2. (an integer times an adjacent integer times 4) + 1
    note: the sum of the two adjacent integers chosen equals the square root
3. (an even integer times an adjacent even integer) + 1
    note: the odd integer between the adjacent even integers is the square root
I'm going to use the latter here.  288/4 = 72  72/2 = 36  36/2 = 18
288/18 = 16      18 - 1 or 16 + 1      17            Also:      
18/2 = 9  72/9 = 8      8 x 9 = 72      8 + 9 = 17
17 is 289's square root.      17 x 17 = 289.
[ sub-note:     5   35   65   95   (x 3)     These are 90-apart locations,
   but 30-apart locations also sometimes contain paired twin primes.
 Both the 30-apart and the 90-apart decades begin at decade 2.
   5 (15/3) is in decade 2 and 95 (285/3) is in decade 29.
   5 x 1 = 5   5 x 7 = 35   5 x 13 = 65   5 x 19 = 95      90-apart
   5 x 1 = 5   5 x 3 = 15  5 x 5 = 25   5 x 7 = 35      30-apart
To get the actual integers/ multiply the products in these
   intentionally-collapsed sequences by "3". ]

Note 2 --  0  4  8   and   2  6  are the two available 
             division-by-4 patterns in each decade

Note 3 --  Regarding the odd squares:  Since the odd squares
             pattern is   1  9  5  9  1   |   1  9  5  9  1, beginning
             at the fifth odd square (81), adjacent occurrences
             of squares having "1" as a final digit/ happen every
             5 squares after 81 and after 121.  Squares having "1"
             as a final digit/ also, starting at 1/ occur every ten odd
             squares (1 x 1 and 19 x 19).  They interlock, however,
             as 19 x 19 is adjacent to 21 x 21.
Integers:   1, 81, 121, 361, 441, 841, 961, 1521, 1681
Decades:   1   9   13    37   45    85   97    153   169
The final-digit-9 squares run:
Integers:   9, 49, 169, 289, 529, 729, 1089, 1369, 1849
Decades:   1   5   17   29    53    73    109   137    185

Note 4 --  final-digit-5 integers evenly divisible by "3"
15   45   75   105   135   165   195   225   255   285   
-
315   345   375   405   435   465   495   525   555   585
-
615   645   675   705   735   765   795   825   855   885
[ sub-note:  The decade pattern for those decades in which "3"
divides evenly into a final-digit-5 integer is:   2  5  8.  
Therefore, only those decades can contain, but may not contain, 
paired twin primes.  One of these decades, 29 (2 + 9 = 11 = 2), 
has 17 x 17 (289) in  it. ]
For the above three rows of integers, the decades are:
2   5   8   11   14   17   20   23   26   29
-
32   35   38   41   44   47   50   53   56   59
-
62   65   68   71   74   77   80   83   86   89

Note 5 --  Whenever "3" divides evenly into a decade, it
also divides evenly into the final-digit-1 integer in that decade.
Some f-d-1 pown squares are in these decades.  One is in 
decade 153:  39 x 39 (1521).  153 digit sums to 9, and so is
not a decade in which "3" divides evenly into its f-d-5 integer.

Note 6 --  More information on decades and "3":
In decade 1, "3" divides evenly into 3 and 6 and 9; and
does the same every third decade from thereon: that is,
it divides evenly into an integer whose final digit is a
3 or 6 or 9.
-
In decade 3, "3" divides evenly into 21 and 24 and 27,
integers whose final digits are 1, 4 and 7 respectively;
and does the same every third decade from thereon.
-
In decade 2, "3" divides evenly into 12 and 15 and 18,
integers whose final digits are 2, 5 and 8 respectively;
and does the same every third decade from thereon.
-
From this it should be obvious why paired twin primes
cannot be found in decades 1, 4, 7, 10, 13, 16, etc.,
or in decades 3, 6, 9, 12, 15, 18, etc.

Note 7 --  Digit addition patterns for integers in decades > 1:
final digit 9:  1 2 3 4 5 6 7 8 9      f-d-1:  2 3 4 5 6 7 8 9 1
f-d-2:  3 4 5 6 7 8 9 1 2      f-d-3:  4 5 6 7 8 9 1 2 3
f-d-4:  5 6 7 8 9 1 2 3 4       f-d-5:  6 7 8 9 1 2 3 4 5
f-d-6:  7 8 9 1 2 3 4 5 6      f-d-7:  8 9 1 2 3 4 5 6 7
f-d-8: 9 1 2 3 4 5 6 7 8      f-d-0:  1 2 3 4 5 6 7 8 9
-
for even squares (if there is a pattern):
4 is 4  16 is 7  36 is 9  64 is 1  100 is 1  144 is 9  196 is 7  
256 is 4  324 is 9   |   400 is 4  484 is 7  576 is 9  676 is 1
784 is 1  900 is 9  1024 is 7  1156 is 4  1296 is 9
4  7  9  1  1  9  7  4  9
-
for odd squares (if there is a pattern):
9 is 9  25 is 7  49 is 4  81 is 9  121 is 4  169 is 7  225 is 9  289 is 1  361 is 1
9  7  4  9  4  7  9  1  1
-
Curious related facts:  17 is 8; 17 x 17 = 289, a 1;  289 is in decade 29, a 2.
39 is 3; 39 x 39 = 1521, a 9; 1521 is in decade 153, a 9.

6:17 PM
Addendum 1:  Began this in Note 3; but moments ago, because I wanted 
to know the changes cycle for odd squares having "1" as a final digit, this:   
1 x 1 = 1      9 x 9 = 81 (9)      11 x 11 = 121 (4)
19 x 19 = 361 (1)      21 x 21 = 441 (9)      29 x 29 = 841 (4)
31 x 31 = 961 (7)      39 x 39 = 1521 (9)      41 x 41 = 1681 (7)
49 x 49 = 2401 (7)      51 x 51 = 2601 (9)      59 x 59 = 3481 (7)
61 x 61 = 3721 (4)      69 x 69 = 4761 (9)      71 x 71 = 5041 (1)
79 x 79 = 6241 (4)      81 x 81 = 6561 (9)      89 x 89 = 7921 (1)
18-term cycle for which the decades are:
1   9   13   37   45   85   |    
97   153   169   241   261   349   |   
373   477   505   625   657   793
Because the final digit for each of these squares is a "1"/
each square/ digit sums to same number its decade does.
This being so, f-d-1 squares do not occur in the decades
where "3" divides evenly into an f-d-5 integer.  Those
decades digit sum to 2, 5 and 8 only.  These decades
digit sum to 1, 4, 7 and 9 only.

Addendum 2:  The f-d-9 squares occur in decades
which digit sum to 1, 2, 5and 8 only, which is why
the 289 square in decade 29 prevents that decade
from containing paired twin primes.
8 PM - This is the 18-term changes cycle for f-d-9 squares:
3 x 3 = 9 (9)    7 x 7 = 49 (4)    13 x 13 = 169 (7)                          1  5  8
17 x 17 = 289 (1)    23 x 23 = 529 (7)    27 x 27 = 729 (9)               2  8  1
33 x 33 = 1089 (9)    37 x 37 = 1369 (1)    43 x 43 = 1849 (4)          1  2  5
47 x 47 = 2209 (4)    53 x 53 = 2809 (1)    57 x 57 = 3249 (9)          5  2  1
63 x 63 = 3969 (9)    67 x 67 = 4489 (7)    73 x 73 = 5329 (1)          1  8  2
77 x 77 = 5929 (7)    83 x 83 = 6889 (4)    87 x 87 = 7569 (9)          8  5  1  


See also:
sw00878math/odd-primes-and-digit-addition
sw00879math/digit-addition-families
sw00881math/pown-and-pewn-squares-notes
sw00882math/descension-by-squares-groups
sw00883math/my-paired-twin-primes-conjecture
sw00884math/more-views-of-my-conjecture  





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Brian A. J. Salchert