I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and there's no prime number $i^{'}<i$ s.t. $n-i^{'}$ is prime. Usually $i$ is much smaller than $n$. By programming, I found some $n$ and $i$. Here are them:
$n(i):6(3), 12(5), 30(7), 98(19), 220(23), 308(31), 556(47), 992(73), 2642(103), 5372(139),7426(173), 43532(211), 54244(233), 63274(293), 113672(313), 128168(331), 194428(359), 194470(383) ,413572(389), 503222(523), 1077422(601), 3526958(727), 3807404(751), 10759922(829), 24106882(929), 27789878(997), 37998938(1039), 60119912(1093) ,113632822(1163), 187852862(1321), 335070838(1427), 419911924(1583) ,721013438(1789), 1847133842(1861).$
All the $n$ and $i$ are the smallest ones. I'm just curious about how fast $n$ increases when $i$ increases.(Under $2\times 10^9$,there's no $n$ for which $i$ is larger than $2000$) .Any comments will be appreciated.
COMMENT.-It is clear that your perspective is the same as solve the equation
$$2n-X=i \space \space \text{where }X\text{ and }\space \space i\text{ are prime and } n, i\space \space \text{are both minimum } $$
I think your problem is very difficult, more than the usual (difficult) problems about prime numbers. What you have on hand is only "brute force" with the limits of computation that this implies. Am I wrong?