Primorial prime

In mathematics, a primorial prime is a prime number of the form pn# ±&thinsp;1, where pn# is the primorial of pn (i.e. the product of the first n primes).

Primality tests show that:


 * pn# −&thinsp;1 is prime for n = 2, 3, 5, 6, 13, 24, ....
 * pn# +&thinsp;1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ....

The first term of the second sequence is 0 because p0# = 1 is the empty product, and thus p0# +&thinsp;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.

, the largest known primorial prime (of the form pn# − 1) is 3267113# −&thinsp;1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.

, 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:


 * Assume that the first n consecutive primes including 2 are the only primes that exist. If either pn# +&thinsp;1 or pn# −&thinsp;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 −&thinsp;1 or 1 when divided by any of the first n primes, and hence all its prime factors are larger than pn).