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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 nth 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 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, ... (OEIS A002110)

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 \(P \# + 1\) 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 Edit

  1. Primorial -- from Wolfram MathWorld

See also Edit

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
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 · Bouncing Factorial