# primorial

We define**(**

*p*#*p*-

**primorial**) to be the product of the primes less than or equal to

*p*. For example:

- 3# = 2
^{.}3 = 6, - 5# = 2
^{.}3^{.}5 = 30, and - 13# = 2
^{.}3^{.}5^{.}7^{.}11^{.}13 = 30030.

*p*-

**prime factorial**. Euclid's proof that there are infinitely many primes provides what may be the first use of

*p*# (the concept, not the notation).

It is customary to only apply the notation *p*# to
primes *p*, but some authors will apply it to any
positive real number (e.g., 10.72# =
2^{.}3^{.}5^{.}7 = 210).
When viewed this way, the function log(*x*#) is
Tschebycheff's function, and the prime number theorem is equivalent to the expression

log(i.e., (logx# ~x,

*x*#)/

*x*approaches 1 as

*x*approaches infinity.)

**See Also:** Factorial, FactorialPrime, MultifactorialPrime

**Related pages** (outside of this work)

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