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?
As mentioned in comments, the value is not always prime, but obviously, any prime factors have to be bigger than $P_k$. Indeed, for the primes $P_k$ after $11$ and less than $50$, the product of the primes up to and including $P_k$, then adding one, results in a value that is not prime, except in the case of $31$.
$$\begin{align} 2\cdot3\cdot5\cdot7\cdot11\cdot13+1 &= 30031 \\&= 59\cdot509\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17+1 &= 510511 \\&= 19\cdot97\cdot277\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19+1 &= 9699691 \\&= 347\cdot27953\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23+1 &= 223092871 \\&= 317\cdot703763\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29+1 &= 6469693231 \\&= 331\cdot571\cdot34231\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37+1 &= 7420738134811 \\&= 181\cdot60611\cdot676421\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41+1 &= 304250263527211 \\&= 61\cdot450451\cdot11072701\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41\cdot43+1 &= 13082761331670031 \\&= 167\cdot78339888213593\\ 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41\cdot43\cdot47+1 &= 614889782588491411 \\&= 953\cdot46727\cdot13808181181\\ \end{align}$$