In Euclid's proof that there are infinitely many primes the number $p_1 p_2 ... p_n + 1$ is constructed and proved to be either a prime, or a product of primes greater than $p_n$.
Trivially, we could also use the number $R_n=p_1 p_2 ... p_n - 1$ to prove the theorem, for n>2.
Intuitively, as $n$ grows, the probability that $R_n$ is prime gets smaller and smaller. Is there a proof that $R_n$ is not prime for any $n$ greater than some integer $M$ ? Or conversely, that there is an infinite number of prime $R_n$ numbers?
A possibly equivalent question: Is there a prime number greater than $p_n$ and smaller than $R_n$ for any $n$ greater than some integer $M$ ?