# Primorial prime

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

Primality tests show that

pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in the OEIS)
pn# + 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 (sequence A228486 in the OEIS)

As of March 2018, the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, found by the PrimeGrid project.[2][3]

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 pn# + 1 or pn# − 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 pn).

## References

1. ^ Weisstein, Eric. "Primorial Prime". MathWorld. Wolfram. Retrieved 18 March 2015.
2. ^ Primegrid.com; forum announcement, 2 March 2011
3. ^ Caldwell, Chris K., The Top Twenty: Primorial (the Prime Pages)
4. ^ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.

• A. Borning, "Some Results for ${\displaystyle k!+1}$  and ${\displaystyle 2\cdot 3\cdot 5\cdot p+1}$ " Math. Comput. 26 (1972): 567–570.