Primorial prime
# 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).

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 $p_0#$ = 1 is the empty product, and thus $p_0#$ + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because $p_1#$ = 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 October 2021, the largest known primorial prime (of the form $p_n#$ − 1) is 3267113# − 1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.

As of 2022, the largest known prime of the form $p_n#$ + 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 $p_n#$ + 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 $p_n).$
