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]

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

As of October 2021[ref], 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.^[2][3]

As of 2022^[update], 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.

• Compositorial
• Euclid number
• Factorial prime

References

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.

This article about a number is a stub. You can help Wikipedia by expanding it.

• v
• t
• e