Primorial

The primorial, a portmanteau of prime and factorial, is formally defined as
\[p_n \# = \prod^{n}_{i = 1} p_i\]
where \(p_n\) is the nth prime.Primorial -- from Wolfram MathWorld
Another slightly more complex definition, which expands the domain of the function beyond prime numbers, is
\[n \# = \prod^{\pi (n)}_{i = 1} p_i\]
where \(p_n\) is the nth prime and \(\pi (n)\) is the prime counting function.
Using either definition, the primorial of n can be informally defined as "the product of all prime numbers up to n, inclusive." For example, \(16 \# = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 = 30,030\).
The primorial's relationship to the Chebyshev function \(\theta (x)\) gives it the property
\[\lim_{n\rightarrow\infty} \sqrt[p_n]{p_n \#} = e\]
where e is the mathematical constant.
The sequence of primorials goes:
1, 2, 6, 30, 210, 2,310, 30,030, 510,510, ... ()
== Euclid's theorem ==
The primorial can be used to prove that there are infinitely many primes (). If there was a largest prime \(P\), then \(P \# + 1\) and \(P \# - 1\) would also be prime, which is a contradiction. Outside of the proof by contradiction, \(p \# + 1\) or \(p \# - 1\) is not always prime. Such primes are called primorial primes.
== Sources ==
== See also ==
Category:Factorials
Category:Functions