# Maximal Goldbach Partitions?

A Goldbach partition $$2n = p + q$$ with $$p$$ and $$q$$ primes and $$p \leqslant q$$ is usually called minimal if the numbers $$2n - k$$ ($$k = 1,\ldots, p-1$$) are all composite.

Reading through the literature, I see that the minimal Goldbach partitions of even integers have been studied a lot (for instance in https://dms.umontreal.ca/~andrew/PDF/Goldbach1.pdf).

Has anybody ever worked on what I'd call maximal Goldbach partitions? By a maximal Goldbach partition I mean a partition $$2n = p + q$$ with $$p$$ and $$q$$ primes, $$p \leqslant q$$ and $$q-p \geqslant 0$$ as small as possible. (E.g. $$13+17$$ would be the maximal Goldbach-partition of $$30$$.)

• You can make realistic conjectures from the random model for the primes : that $X_n = 1_{n \text{ is prime}}$ is a sequence of independent random variables with $P(X_n=1)= 1/\log n$ – reuns Oct 17 '18 at 22:27