Posting in MO since it was unanswered in MSE.
Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 = 3 + 17 = 7 + 13$ if we start searching from the smallest prime we will encounter the pair $(3,17)$ first and say that $20$ satisfied Goldbach's conjecture. We do not have to go all the way to the pair $(7,13)$.
Definition 1: The smallest Goldbach prime of an even number $n$ is defined as the pair $(p_n,n-p_n)$ where $p_n$ is the smallest prime such that $n-p_n$ is also a prime.
I found that $p_n$ is actually much smaller than $n$. Experimental data for $n \le 230751000000 $ suggests an asymptotic relation of the form
Question: Let $n$ be an even number which can be represented as the sum of two primes and $p_n$ its smallest Goldbach's prime. What is the limiting value of
$$ \frac{1}{x}\sum_{n \le x} p_n - \log x $$
Experimental data for $x \le 2.25 \times 10^{11}$ gives a value of about $1.313$ and it decreases slowly.
Additional data: Given $n$, we can ask how large can $p_n$. For even $n \le 3.6 \times 10^{10}$, the largest value of $p_n$ occurs as the following values of $n$ as shown below by maximal $(n, p_n)$ pairs.
(n,p_n)
(4,2)
(6,3)
(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)
(7473202036,1877)
(11001080372,1879)
(12703943222,2029)
(21248558888,2089)
(35884080836,2803)
