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 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 (sequence A228486 in the OEIS).

As of September 2024, the largest known primorial prime (of the form p_n# − 1) is 4778027# − 1 (n = 334,023) with 2,073,926 digits, found by the PrimeGrid project.

As of September 2024, the largest known prime of the form p_n# + 1 is 5256037# + 1 (n = 365,071) with 2,281,955 digits, found in 2024 by PrimeGrid.

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).

See also

• Compositorial
• Euclid number
• Factorial prime

See also

• A. Borning, "Some Results for $k!+1$ and $2\cdot 3\cdot 5\cdot p+1$" Math. Comput. 26 (1972): 567–570.
• Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
• Harvey Dubner, "Factorial and Primorial Primes." J. Rec. Math. 19 (1987): 197–203.
• Paulo Ribenboim, The New Book of Prime Number Records. New York: Springer-Verlag (1989): 4.