By analogy with the definition of the factorial function, the primorial of a prime number p (often denoted by a hash symbol: p#) is defined as the product of all primes up to and including p. For example,

7# = 2 x 3 x 5 x 7 = 210

Mathematicians (with perhaps more computer time than they deserve) have searched out many huge primes which are one greater or less than various primorials. This is reminiscent of the use of primorials in Euclid's proof of the infinitude of primes.

An interesting conjecture put forward by Odlyzko, Rubinstein and Wolf in Experimental Mathematics (Vol. 8, No. 2, 1999) states that the set of primorials, along with the number 4, is precisely the set of jumping champions.

While it appears that n# is a trivial thing to consider, nevertheless it has some bizarre (and unlooked-for) properties, such as the following (from

 lim (p(n)#)       = e

In the above relation, p(n) is the largest prime less than or equal to n, and e is Napier's number (which is more commonly known as Euler's number).

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