Definitions

# Primorial prime

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

pn# is the primorial of pn.

pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ...

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

The first few primorial primes are

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209
, the largest known primorial prime is 392113#+1 with 169966 digits, found in 2001 by Daniel Heuer.

It is widely believed, but false, that 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 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).

In fact, Euclid's proof did not assume that a finite set contains all primes that exist. Rather, it said: consider any finite set of primes (not necessarily the first n primes; e.g. it could have been the set {3, 11, 47}), and then went on from there to the conclusion that at least one prime exists that is not in that set.

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