In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).
Primality tests show that
The first term of the second sequence is 0 because p0# = 1 is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because p1# = 2, and 2 − 1 = 1 is not prime.
The first few primorial primes are
As of October 2021[ref], the largest known primorial prime (of the form pn# − 1) is 3267113# − 1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.
As of 2022[update], the largest known prime of the form pn# + 1 is 392113# + 1 (n = 33,237) with 169,966 digits, found in 2001 by Daniel Heuer.
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: