[1] Primality tests show that: The first term of the third sequence is 0 because p0# = 1 (we also let p0 = 1, see Prime_number#Primality_of_one, hence the first term of the fourth sequence is 1) is the empty product, and thus p0# + 1 = 2, which is prime.
The first few primorial primes are 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (sequence A228486 in the OEIS).
As of December 2024[ref], the largest known prime of the form pn# − 1 is 6533299# − 1 (n = 446,895) with 2,835,864 digits, found by the PrimeGrid project.
As of December 2024[update], the largest known prime of the form pn# + 1 is 7351117# + 1 (n = 498,865) with 3,191,401 digits, also found by the PrimeGrid project.
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:[2] This article about a number is a stub.