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).^[1]

Primorial prime

No. of known terms	51
Conjectured no. of terms	Infinite
Subsequence of	p# ± 1
First terms	2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309
Largest known term	7351117! + 1
OEIS index	A228486

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).

See also

• Compositorial
• Euclid number
• Factorial prime

References

1. Weisstein, Eric. "Primorial Prime". MathWorld. Wolfram. Retrieved 18 March 2015.
2. Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.

See also

• 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.