I know that the method of 'factorial' can guarantee that there are arbitrarily large gaps between consecutive primes, but I have a question about this method, consider the number $$g=p_1\cdot p_2...p_n$$
then we know that $g+2$ is not a prime because it's a multiple of two, and the same thing with 3, and so on, but near the end, we would have this $$g+p_n \text{ a multiple of }p_n$$ $$g+(p_n+1) \text{ an even number }$$
so what about $g+p_n+2$, is this the last number we would have before getting a number that we're not sure whether it's composite or not? note that $p_n,p_n+1,p_n+2$ are three consecutive numbers so one of them is divisible by $3$, but this doesn't implies that $p_n+2$ is divisible, this thing depends whether $p_n\equiv 1\pmod 3$ or not.