Primorial prime

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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 28 February 2012 (2012 -02-28)^[ref], the largest known primorial prime is 1098133# − 1 with 476,311 digits, found by the PrimeGrid project.^[1]

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.^[2]

See also

• Primorial
• Factorial prime
• Euclid number
• PrimeGrid

References

• A. Borning, "Some Results for and " 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.

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