Consider for , for prime, . Consider for , for prime, . Due to errors in the statement, a couple of examples: , we take both the smaller of the two primes as well as the smallest prime of all such pairs. , so we take and .
Find the limit and .
Anwer from JoeBlow (to the first one, slightly changed due to Latex syntax rendering problem):
So you’re asking for where is the smallest prime such that is prime (it goes without saying we are assuming the Goldbach conjecture — or at least some weaker asymptotic version of it — throughout this discussion).
If this limit were not , it would mean we can find arbitrarily large such that is not prime for all primes for some . If we pretend that primes are randomly distributed in with a probability distribution consistent with PNT, the probability of such an event for a fixed is bounded roughly by . But Borel-Cantelli then implies that the event occurs for only finitely many with probability .
Of course the probabilistic argument doesn’t quite hold water, but it suggests heuristically that it would be extremely surprising if this limit weren’t — as surprising as Goldbach failing infinitely many times (because in this probabilstic framework the event has the same order of magnitude as the event “the Goldbach conjecture fails for “).