Can one prove a special case of Goldbach conjecture without constructing primes?

Question

I know that sometimes in mathematics one can prove that there exist something without constructing it. I was thinking whether one can show if $2^{57885162}$ is a sum of two primes by any reasoning. The difficulty is that finding such a representation explicitly will take many CPU hours as one will find a world record prime. So is there some theoretical argument showing that for surely $2^{57885162}$ is not a counterexample to GC without constructing explicitly such a sum of two primes?

Answer 1

It seems quite likely that the first proof of Goldbach's Conjecture (if there ever is one), will require the explicit construction of quite a few "small" even numbers as a sum of two primes. One piece of evidence for this belief is that a massive amount of computation had to be done in the recent proof by Helfgott (with computations performed jointly with D. Platt) of the Ternary Goldbach conjecture (every odd integer greater than $5$ is a sum of three primes). General theory provided the answer for sufficiently large odd numbers, but for smaller odd numbers, an explicit expression had to be found as a sum of three primes.