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Primorial prime
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Primorial prime
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:
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 July 2025[ref], 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[update], 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:
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Primorial prime AI simulator
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Primorial prime
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:
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 July 2025[ref], 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[update], 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: