Let $P_k$ be a prime number and let $P = 2.3.\dots.P_k$. be the product of all primes smaller or equal to $P_k$.
Then $P+1$ is either a prime number or not. If it is not a prime number it shares factors with $P$. Let the product of these factors be $P_r$ such that $P=P_sP_r$.
Then we can write $P+1 = P_rm$ where $m$ is some product of prime numbers. From this it follows that $P_r(P_s -m) = -1$.
Now here's where I get shaky. Does this not imply that $P_r=1$. Does this then not mean that $P+1$ does not in fact share any factors with $P$ and that therefore it is a prime number itself?
Of course this cannot be a prime number generator but where did I make the error?