Primorial prime

In mathematics, a primorial prime is a prime number of the form p_n# ± 1, where p_n# is the primorial of p_n (i.e. the product of the first n primes).^[1]

Primality tests show that:

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

p_n# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ... (sequence A014545 in the OEIS).

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 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (sequence A228486 in the OEIS).

As of September 2024^[ref], the largest known primorial prime (of the form p_n# − 1) is 4778027# − 1 (n = 334,023) with 2,073,926 digits, found by the PrimeGrid project.^[2][3]

As of September 2024^[update], the largest known prime of the form p_n# + 1 is 5256037# + 1 (n = 365,071) with 2,281,955 digits, found in 2024 by PrimeGrid.

Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:^[4]

Assume that the first n consecutive primes including 2 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; 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 all its prime factors are larger than p_n).