I am looking at the Goldbach Conjecture because I think it's interesting. One thing I've noticed using brute force programming methods is this:
It seems up, at least up to numbers with 6 or 7 digits, that given two primes are a "Goldbach Pair", the average relative distance of these two numbers from the number N/2 appears to be N/4.
Let me rephrase this a few ways for clarity, and provide some examples.
Consider the Goldbach pairs for the number (N=8). They are (3,5), and that's it. There's only one pair for (N=8). The numbers "3" and "5" are both (N/2)+1/4*(N/2) and (N/2)-1/4*(N/2). Consider the Goldbach pairs for the number (N=10). They are (3,7) and (5,5). For the purposes of what I am saying here, discount all instances where the Goldbach pairs consist of two of the same numbers. So exclude (5,5) from what I am saying. The numbers "3" and "7" are both (N/2)+2/5*(N/2) and (N/2)-2/5*(N/2).
All Goldbach pairs will be (N/2)+C*(N/2) and (N/2)-C*(N/2)
What I am finding using my program is the results for C seem to tend toward .5 or 1/2 "on average".
Is there a proof that C must be this way? Is there a good explanation for this?
Basically what it seems is that given a random Goldbach pair, it has the same probability of being 1 away from a given N/2 as it does of being any other distance from N/2, which also seems relevant to the twin primes conjecture.