Primorial prime

In mathematics, primorial primes are prime numbers of the form p_n# ± 1, where p_n# is the primorial of p_n (that is, the product of the first n primes).

It follows that

p_n# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in OEIS)

p_n# + 1 is prime for n = 1, 2, 3, 4, 5, 11, ... (sequence A014545 in OEIS)

The first few primorial primes are

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209

As of 2010, the largest known primorial prime is 843301# − 1 with 365,851 digits, found in 2010 by the PrimeGrid project.

It is widely believed, but false, that the idea of primorial primes appears in Euclid's proof of the infinitude of the prime numbers: First, assume that the first n primes 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; note that 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 cannot be a multiple of any of them).

In fact, Euclid's proof did not assume that a finite set contains all primes that exist. Rather, it said: consider any finite set of primes (not necessarily the first n primes; e.g. it could have been the set {3, 11, 47}), and then went on from there to the conclusion that at least one prime exists that is not in that set.

See also

• Primorial
• Factorial prime
• Euclid number
• PrimeGrid

References

• A. Borning, "Some Results for k! + 1 and $2 \cdot 3 \cdot 5 \cdot p + 1$" Math. Comput. 26 (1972): 567–570.
• Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
• Weisstein, Eric W., "Primorial Prime" from MathWorld.
• 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.