Mind you P's and Q's

Goldbach's conjecture would have that every even integer $\gt 2$ can be expressed as the sum of two primes. Now while this conjecture has not been proved, it has been confirmed up to $4 \times 10^{18}$ , so we don't need to be looking for even numbers just yet, including $2 = 5 - 3$ . Now every odd integer up to $21$ is $\pm 2$ with respect to a prime, but not $23$ , as neither $21$ nor $25$ is prime. Since the sum of two odd primes is always even, we can conclude that $P = 23$ , notably the first odd non-twin prime.

Now for $Q$ we are again not looking for even numbers just yet. For odd integers, unless a prime gap exceeds $6$ , any odd composite will be within $2$ of a prime, so we are looking for the first prime gap that is $8$ or more. This occurs between $89$ and $97$ , with the odd composite $93 = 3 \times 31$ differing by $4$ from each of its prime neighbors. Thus $Q = 93$ , and so $Q - P = 93 = 23 = \boxed{70}$ .