What is the difference?

Assume Goldbach's conjecture , which states that every even integer $>2$ can be written as the sum of two primes (which remains unproven to this date).

Let the minimum difference between $p$ and $q$ be $d$ .

Now, $d$ can not be $1$ , since there would be no natural number between $p$ and $q$ . $d$ can not be $2$ , because the only number between $p$ and $q$ would be even. The difference can not be an odd number since there exist no consecutive primes with an odd difference $>1$ , as one of them would have to be even. $d$ can not be $4$ since the only odd number between $p$ and $q$ would have a difference of $2$ to $p$ and $q$ . $d$ can also not be $6$ since there would be $2$ odd numbers between $p$ and $q$ , each of them having a distance of $2$ to the nearest prime.

In the case $d=8$ , there exists an odd number between $p$ and $q$ which does not have a difference of $2$ to either $p$ or $q$ and can thus not be written as the sum or difference of $2$ primes. Thus $d=\boxed{8}$ .