TITLE: Is there an infinite number of primes constructed as in Euclid's proof?
QUESTION [20 upvotes]: 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. 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 $M$ ?

REPLY [3 votes]: Euclid's proof says that if there were only finitely many primes, the product of these primes +1 would be a new prime. It arrives at a contradiction.  
Hence, it really makes no assertion about the product of the first $n$ primes, plus one.