Primorial

For the $i$th Prime $p_i$,

$$\mathop{\rm primorial}(p_i)=p_i\char93 \equiv\prod_{j=1}^i p_j.$$

The first few values for $p_i\char93$, are 2, 6, 30, 210, 2310, 30030, 510510, ... (Sloane's A002110).

$p\char93 -1$ is Prime for Primes $p=3$, 5, 11, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, ... (Sloane's A014563; Guy 1994), or $p_n$ for $n=2$, 3, 5, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, .... $p\char93 +1$ is known to be Prime for the Primes $p=2$, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, ... (Sloane's A005234; Guy 1994, Mudge 1997), or $p_n$ for $n=1$, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, ... (Sloane's A014545). Both forms have been tested to $p=25000$ (Caldwell 1995). It is not known if there are an infinite number of Primes for which $p\char93 +1$ is Prime or Composite (Ribenboim 1989).

See also Factorial, Fortunate Prime, Prime Sum Smarandache Near-to-Primorial Function, Twin Peaks

References

Borning, A. ``Some Results for $k!+1$ and $2\cdot 3\cdot 5\cdot p+1$.'' Math. Comput. 26, 567-570, 1972.

Buhler, J. P.; Crandall, R. E.; and Penk, M. A. ``Primes of the form $M!+1$ and $\cdot 3\cdot 5\cdot p+1$.'' Math. Comput. 38, 639-643, 1982.

Caldwell, C. ``On The Primality of $n!\pm 1$ and $2\cdot 3\cdot 5\cdots p\pm 1$.'' Math. Comput. 64, 889-890, 1995.

Dubner, H. ``Factorial and Primorial Primes.'' J. Rec. Math. 19, 197-203, 1987.

Dubner, H. ``A New Primorial Prime.'' J. Rec. Math. 21, 276, 1989.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 7-8, 1994.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/primorial-.Z and ftp://sable.ox.ac.uk/pub/math/factors/primorial+.Z.

Mudge, M. ``Not Numerology but Numeralogy!'' Personal Computer World, 279-280, 1997.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 4, 1989.

Sloane, N. J. A. Sequences A002110/M1691, A005234/M0669, A014545, and A014563 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Temper, M. ``On the Primality of $k!+1$ and $\cdot 3\cdot 5\cdots p+1$.'' Math. Comput. 34, 303-304, 1980.