Primorial prime

Primorial prime
No. of known terms52
Conjectured no. of termsInfinite
Subsequence ofp# ± 1
First terms2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309
Largest known term9562633# + 1
OEIS indexA228486

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

Primality tests show that:

pn# − 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). (pn = 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))
pn# + 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, 637491, ... (sequence A014545 in the OEIS). (pn = 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, 9562633, ..., (sequence A005234 in the OEIS))

The first term of the third sequence is 0 because p0# = 1 (we also let p0 = 1, see 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 2025, it is not known whether there are infinitely many primorial primes, and it is also not known whether infinitely many numbers of the form pn# ± 1 are composite numbers.

As of July 2025, the largest known prime of the form pn# − 1 is 6533299# − 1 (n = 446,895) with 2,835,864 digits, found by the PrimeGrid project.

As of July 2025, the largest known prime of the form pn# + 1 is 9562633# + 1 (n = 637,491) with 4,151,498 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:

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).

See also

References

  1. ↑ Weisstein, Eric. "Primorial Prime". MathWorld. Wolfram. Retrieved 18 March 2015.
  2. ↑ Weisstein, Eric W. "Primorial Prime". MathWorld. Retrieved 2025-11-29.
  3. 1 2 https://t5k.org/top20/page.php?id=5#records
  4. ↑ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.

See also