# Primorial

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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 *n*th prime.^{[1]}

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 *n*th 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, \(16 \# = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 = 30030\).

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.

### Euclid's theorem Edit

The primorial can be used to prove that there are infinitely many primes (Euclid's theorem). If there was a largest prime \(P\), then \(P \# + 1\) would also be prime, which is a contradiction. Outside of the proof by contradiction, \(p \# + 1\) is not always prime.

### Sources Edit

### See also Edit

**Multifactorials:**Double factorial · Multifactorial

**Falling and rising:**Falling factorial · Rising factorial

**Other mathematical variants:**Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial

**Tetrational growth:**Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover

**Array-based extensions:**Hyperfactorial array notation · Nested factorial notation

**Other googological variants:**· Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial