Primorial prime

From Exampleproblems

In mathematics, primorial primes are prime numbers of the form p_n# ± 1, where:

p_n# is the primorial of p_n.

p_n# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (Sloane A057704)

p_n# + 1 is prime for n = 1, 2, 3, 4, 5, 11, ... (Sloane A014545)

As of 2005, the largest known primorial prime is 392113#+1, found in 2001 by Daniel Heuer.

The idea of primorial primes appears in Euclid's proof of the infinitude of the prime numbers: First, assume that the first n primes are the only primes that exist. If either p_n# + 1 or p_n# - 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; note that each of these two numbers has a remainder of either p−1 or 1 when divided by any of the first n primes, and hence cannot be a multiple of any of them).