Primorial prime

In mathematics, primorial primes are prime numbers of the form "p_n"# ± 1, where:

: "p_n"# is the primorial of "p_n".

: "p_n"# − 1 is prime for "n" = 2, 3, 5, 6, 13, 24, ... OEIS|id=A057704

: "p_n"# + 1 is prime for "n" = 1, 2, 3, 4, 5, 11, ... (OEIS2C|id=A014545)

The first few primorial primes are

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

As of|2008|url=http://primes.utm.edu/top20/page.php?id=5, the largest known primorial prime is 392113#+1 with 169966 digits, found in 2001 by Daniel Heuer.

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 "n"th 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. [http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html]

See also

* Primorial
* Factorial prime
* Euclid number

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, [http://primes.utm.edu/top20/page.php?id=5 "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.