Primorial prime
In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn.
Primality tests show that:
The first term of the third 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.
, it is not known whether there are infinitely many primorial primes, and it is also not known whether infinitely many numbers of the form pn# ± 1 are composite numbers.
, the largest known prime of the form pn# − 1 is 6533299# − 1 with 2,835,864 digits, found by the PrimeGrid project.
, the largest known prime of the form pn# + 1 is 9562633# + 1 with 4,151,498 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: