Is the Goldbach Conjecture part of a larger pattern?

[raycb]

I wonder if the Goldbach Conjecture could be extended, so that the even number halfway between two primes that differ by 4n + 2 can be written as the sum of two primes from pairs that differ by 4n + 2? This conjecture often fails for the first few numbers, but then seems to begin working.

For example, the midpoint of 3 and 13 is 8, which cannot be written as the sum of two primes from pairs that differ by 10, but the midpoint of 103 and 113 is 108, which can be written as the sum of 29 and 89, which come from the pairs 19 and 29, and 79 and 89.

 

[CRGreathouse]

I wonder if the Goldbach Conjecture could be extended, so that the even number halfway between two primes that differ by 4n + 2 can be written as the sum of two primes from pairs that differ by 4n + 2? This conjecture often fails for the first few numbers, but then seems to begin working.

For example, the midpoint of 3 and 13 is 8, which cannot be written as the sum of two primes from pairs that differ by 10, but the midpoint of 103 and 113 is 108, which can be written as the sum of 29 and 89, which come from the pairs 19 and 29, and 79 and 89.

Your question seems to be:

Is it true that for every pair of odd primes p < q with q - p = 4n + 2 that there are r,s with p + 2n + 1 = r + s, where r, s, r + 4n + 2, and one of s ± (4n + 2) are prime?​

or rather whether this is true for large enough p+q.

I find lots of exceptions to this: 48 exceptions with p < q < 100, 1805 exceptions up with p < q < 1000, and 77732 exceptions with p < q < 10000.

Testing code:

test(p,q)=my(diff=q-p,m=p+diff/2);forprime(r=2,m\2,if(isprime(m-r)&isprime(r+diff)&(isprime(m-r+diff)|isprime(m-r-diff)),return(r)));0
upto(lim)=my(s=0);forprime(p=3,lim,forprime(q=p+2,lim,if((q-p)%4==0,next);if(!test(p,q),print1([p,q]);s++)));s