## Goldbach's Conjecture, another slant

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• Hello all Bill s comments led me to an interesting view of Goldbach s conjecture. GB conjecture: Every even number 3 is the sum of two primes at least once.
Message 1 of 1 , Nov 26, 2001
Hello all

Bill's comments led me to an interesting view of Goldbach's
conjecture.

GB conjecture: Every even number > 3 is the sum of two primes at
least once.

For instance, 26 = 13+13 = 7+19 = 3+23

That is, 26 = (13-0)+(13+0) = (13-6)+(13+6) = (13-10)+(13+10)

In shorter notation, 13: (0,6,10)

Restating the GB conjecture: Every number n > 1 is exactly between
two primes, at least once.

Here is a list of n up to 32:

2: (0) 17: (0,6,12,14)
3: (0) 18: (1,5,11,13)
4: (1) 19: (0,12)
5: (0,2) 20: (3,9,17)
6: (1) 21: (2,8,10,16)
7: (0,4) 22: (9,15,19)
8: (3,5) 23: (0,6,18,20)
9: (2,4) 24: (5,7,13,17,19)
10: (3,7) 25: (6,12,18,22)
11: (0,6,8) 26: (3,15,21)
12: (1,5,7) 27: (4,10,14,16,20)
13: (0,6,10) 28: (9,15,25)
14: (3,9) 29: (0,12,18,24)
15: (2,4,8) 30: (1,7,11,13,17,23)
16: (3,13) 31: (0,12,28)
17: (0,6,12,14) 32: (9,15,21,29)

Look at where, say, the number 3 appears throughout this list in the
brackets. It is at n=8,n=10,n=14,n=16,n=20 and n=26. Subtract 3 from
each of these numbers and you get 5,7,11,13,17,23. Add 3 to each of
these numbers and you get 11,13,17,19,23,29.

Similarly look at where the number 5 appears throughout this list in
the brackets. It is at n=8,n=12,n=18,n=24. Subract 5 from each of
these numbers and you get 3,7,13,19. Add 5 to each of these numbers
and you get 13,17,23,29.

The number 6 appears at n=11,n=13,n=17,n=23,n=25. Subtract 6 from
each of these numbers and you get 5,7,11,17,19. Add 6 to each of
these numbers and you get 17,19,23,29,31.

It stunned me a little at first to see this, but it is an entirely
expected result, really! :)

The number zero occurs with the frequency of the primes (of course).
The number one occurs with the frequency of the twin primes (of
course). A perhaps a not so obvious proposal is that the numbers
2,4,8,16, ect. will also each occur at the frequency of the twin
primes. The numbers 3,6,9,12,18,24,27, ect. will each occur at the
frequency of twice the twin primes. The numbers 5,10,20,25,50 ect.
will each occur at the frequency of 4/3 the twin primes.

Well enough silliness for now

Mark
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