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).^[1]

Primality tests show that:

p_n# − 1 is prime for n = 2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, 334023, 435582, 446895, ... (sequence A057704 in the OEIS). (p_n = 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299, ... (sequence A006794 in the OEIS))

p_n# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, ... (sequence A014545 in the OEIS). (p_n = 1, 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, ..., (sequence A005234 in the OEIS))

The first term of the third sequence is 0 because p₀# = 1 (we also let p₀ = 1, see Prime_number#Primality_of_one, hence the first term of the fourth sequence is 1) is the empty product, and thus p₀# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1 (hence the first term of the second sequence is also not 2), 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 December 2024^[ref], the largest known prime of the form p_n# − 1 is 6533299# − 1 (n = 446,895) with 2,835,864 digits, found by the PrimeGrid project.

As of December 2024^[update], the largest known prime of the form p_n# + 1 is 7351117# + 1 (n = 498,865) with 3,191,401 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:^[2]

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).

• Compositorial
• Euclid number
• Factorial prime

• A. Borning, "Some Results for ${\displaystyle k!+1}$ and ${\displaystyle 2\cdot 3\cdot 5\cdot p+1}$" 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.

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