By last, I mean the most recently discovered prime number. What was the length of time between the discovery of the last two prime numbers?
Apart from very large "special" primes, it is also possible to make/construct large "certified" primes, using the Pocklington-Lehmer criterion. That is, although one must "search" ("randomly") for primes readily certifiable, once they are found one can "attach" to them a small amount of data that anyone interested could use to verify their primality (via Pocklington-Lehmer, for example). In particular, although the search is obviously probabilistic, the certification is not.