I've been trying to visualize the problem in many different ways, tables and graphs and geometry and anything else that seems plausible. Here's one I nailed down this morning that's concise and readily graspable for any coder.

First, ignore the 2. It just gets in the way.

And then we get going. Starting with 3, we'll build up a list of primes and I'll show how you can easily check for Goldbach numbers while you do. As you build a string of bits, on each new number, you reverse the bits you have so far and slide it over one.

Much easier understood with a demonstration, starting with n=6 => n/2=3 and incrementing n=n+2, as it must be even.

``````n=6   3
prm?  1
rev   1

n=8   3 5
prm?  1 1
rev   1 1

n=10  3 5 7
prm?  1 1 1
rev   1 1 1

n=12  3 5 7 9
prm?  1 1 1 0
rev   0 1 1 1

n=14  3 5 7 9 11
prm?  1 1 1 0 1
rev   1 0 1 1 1

n=16  3 5 7 9 11 13
prm?  1 1 1 0  1  1
rev   1 1 0 1  1  1

n=18  3 5 7 9 11 13 15
prm?  1 1 1 0  1  1  0
rev   0 1 1 0  1  1  1

n=20  3 5 7 9 11 13 15 17
prm?  1 1 1 0  1  1  0  1
rev   1 0 1 1  0  1  1  1
``````

All we're doing with rev is reversing the order of the "prime?" bits. If you watch from one to the next, you can see that this is actually resulting in a kind of ongoing shift each time. Observe:

``````prm?   1  1  1  0  1  1  0  1  1  0  1  0  0  1
n      3  5  7  9 11 13 15 17 19 21 23 25 27 29
6      1
8      1  1
10     1  1  1
12     0  1  1  1
14     1  0  1  1  1
16     1  1  0  1  1  1
18     0  1  1  0  1  1  1
20     1  0  1  1  0  1  1  1
22     1  1  0  1  1  0  1  1  1
24     0  1  1  0  1  1  0  1  1  1
26     1  0  1  1  0  1  1  0  1  1  1
28     0  1  0  1  1  0  1  1  0  1  1  1
30     0  0  1  0  1  1  0  1  1  0  1  1  1
32     1  0  0  1  0  1  1  0  1  1  0  1  1  1
``````

Notice all we are doing, for each new line, is taking prm?(n), sticking it at the beginning of the new line, and shoving over all the rest. Same as above, but easier to see here. Note also the prime bits read down vertically in the regular order.

###### So?

The thing is, you can check for any and all Goldbach numbers by simply bitwise ANDing two corresponding strings together. (This may actually be an efficient way to code this, should such a need arise.) Take n=28:

``````       3  5  7  9 11 13 15 17 19 21 23 25
prm?   1  1  1  0  1  1  0  1  1  0  1  0
rev    0  1  0  1  1  0  1  1  0  1  1  1
-----------------------------------------
&    0  1  0  0  1  0  0  1  0  0  0  1
``````

The resulting bits identify which numbers are "Goldbach numbers," primes which will sum to n. In this case we get two pairs:

5 + 23 = 28
11 + 17 = 28

This process will give you all valid Goldbach numbers for a given n.

``````  prm? 1  1  1  0  1  1  0  1  1  0  1  0  0  1
n      3  5  7  9 11 13 15 17 19 21 23 25 27 29
6     (1)                                            3+3=6
8     (1  1)                                         3+5=8
10    (1 (1) 1)                                      3+7=10   5+5=10
12     0 (1  1) 1                                    5+7=12
14    (1  0 (1) 1  1)                                3+11=14  7+7=14
16    (1 (1  0  1  1) 1)                             3+13=16  5+11=16
18     0 (1 (1  0  1) 1) 1                           5+13=18  7+11=18
``````

...and so on. I added the parentheses to emphasize the pairs, but it's just matching up by working from the outsides to the center. It did make me notice something I should have seen before, which is that if you don't want to bother with all the reversing and stuff, you accomplish the same thing by taking the prime bit string, counting in from both sides, and watching for matches that are both 1. Same exact thing.

The astute observer may point out that I'm not actually doing anything in this write-up, which is true. Especially viewed this most recent way, all we're doing is literally taking the Goldbach Conjecture by its definition, and looking manually for pairs of primes that add up. If there is something to be gained from this approach, I suspect it lies in study of the "sliding" nature of the reversal version, and seeing how it conspires not to leave holes.

This is almost certainly biased on my hunch that Goldbach's undoing lies in a thorough examination of the mechanism driving its avoidance of symmetry; to violate Goldbach, there would have to be an instance of a certain symmetry, which would in turn be representative of a periodic pattern to a degree verboten to primes.

But hey, that's just me.