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 $30$.)

  • $\begingroup$ 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 $P(X_n=1)= 1/\log n$ $\endgroup$ – reuns Oct 17 '18 at 22:27

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