# Primorial prime

In mathematics, primorial primes are prime numbers of the form pn# ± 1, where pn# is the primorial of pn (that is, the product of the first n primes). [1]

According to this definition,

pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in OEIS)
pn# + 1 is prime for n = 1, 2, 3, 4, 5, 11, ... (sequence A014545 in OEIS)

The first few primorial primes are

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209

As of 28 February 2012, the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, found by the PrimeGrid project.[2]

Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: [3]

Assume that the first n consecutive primes including 2 are the only primes that exist. If either pn# + 1 or pn# − 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).

• A. Borning, "Some Results for $k! + 1$ and $2 \cdot 3 \cdot 5 \cdot p + 1$" Math. Comput. 26 (1972): 567–570.