Primorial prime

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

According to this definition,

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, as 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

As of 28 February 2012, the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, 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:

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).