sw00890math--Goldbach-conjecture-fiddlings

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  Every even number is the sum of two primes.

 Since "2" is a prime number, "4" is the first even number, and since
2 + 2 = 4 and 3 + 3 = 6 is allowed, that is how my fiddlings will be.
PM 9:01 - Have already changed my mind on this.  3 + 5 = 8 and 
5 + 7 = 12 are the only solutions for 8 and 12. 

 3 + 7 = 10        5 + 7 = 12       3 + 11 = 14     3 + 13 = 16      7 + 11 = 18
7 + 13 = 20       3 + 19 = 22    11 + 13 = 24     3 + 23 = 26     11 + 17 = 28
13 + 17 = 30    13 + 19 = 32    11 + 23 = 34    17 + 19 = 36    19 + 19 = 38
17 + 23 = 40    19 + 23 = 42    13 + 31 = 44    17 + 29 = 46    19 + 29 = 48
13 + 37 = 50    23 + 29 = 52    23 + 31 = 54    13 + 43 = 56    29 + 29 = 58
17 + 43 = 60    19 + 43 = 62    23 + 41 = 64    13 + 53 = 66    31 + 37 = 68
23 + 47 = 70    29 + 43 = 72    31 + 43 = 74    17 + 59 = 76    31 + 47 = 78
37 + 43 = 80    23 + 59 = 82    41 + 43 = 84     7 + 79 = 86     41 + 47 = 88
43 + 47 = 90    19 + 73 = 92    41 + 53 = 94    13 + 83 = 96    37 + 61 = 98
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0 pattern 10 through 100:   3 7   7 3   3 7   7 3   3 7   7 3   3 7   7 3  3 7   7 3   
2, 4, 6, 8 patterns not yet defined, but there are final-digit limitations.  At 112/
41 + 71 or 29 + 83, the point being that both combinations work with f-d-2 sums.
At 114/ 53 + 61 and 7 + 107 work.  At 116/ 3 + 113 and 7 + 109 work.
At 118/ 17 + 101 and 29 + 89 work.  118 is also 59 + 59.  As to 0, the pattern
I have shown is not the only possible one since f-d-1 and f-d-9 sum to 0, but it
is the only possible one--if 5 + 5 is excluded--before 11 + 19. 
-
47 + 53 = 100   43 + 59 = 102   43 + 61 = 104   23 + 73 = 106   47 + 61 = 108
43 + 67 = 110 

AM 9:27 5-19-2008 - The above merely begins to prove Goldbach's conjecture,
and I actually don't think that is why I am doing this.  Curiosity is the real reason.
However, this fiddling might be a backdoor way of uncovering a prime number 
fact or two through final digits.

Consider these:  1 and 9 can work for 0 here and there, but 3 and 7 might work
                          for 0 infinitely.
5 and 7 work for 2 once only and where no other primes can, but otherwise 1 
and 1 works for 2, and 3 and 9 might work for 2 consistently.
2 and 2 works for 4 once only and where no other primes can, but otherwise 7
and 7 works for 4, and 1 and 3 work for 4.
3 and 3 works for 6 where it alone can work, but also beyond, and 7 and 9
also work for 6.
3 and 5 work for 8 where they alone can work, but beyond "8" 1 and 7 work
along with 9 and 9.
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I think these limits are meaningful.          




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Brian A. J. Salchert