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{{Distinguish|primordial (disambiguation)}}
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[[Image:Primorial pn plot.png|thumb|300px|''p<sub>n</sub>''# as a function of ''n'', plotted logarithmically.]]
[[Image:Primorial n plot.png|thumb|300px|''n''# as a function of ''n'' (red dots), compared to ''n''!. Both plots are logarithmic.]]


In [[mathematics]], and more particularly in [[number theory]], '''primorial''' is a function from [[natural number]]s to natural numbers similar to the [[factorial]] function, but rather than successively multiplying positive integers, only [[prime number]]s are multiplied.
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There are two conflicting definitions that differ in the interpretation of the argument: the first interprets the argument as an index into the sequence of prime numbers (so that the function is [[strictly increasing]]), while the second interprets the argument as a bound on the prime numbers to be multiplied (so that the function value at any composite number is the same as at its predecessor).
 
The name "primorial", attributed to [[Harvey Dubner]], draws an analogy to ''primes'' the same way the name "factorial" relates to ''factors''.
 
== Definition for prime numbers ==
 
For the ''n''th prime number ''p<sub>n</sub>'' the primorial ''p<sub>n</sub>#'' is defined as the product of the first ''n'' primes:<ref name="mathworld">{{Mathworld | urlname=Primorial | title=Primorial}}</ref><ref name="OEIS A002110">{{OEIS|id=A002110}}</ref>
 
:<math>p_n\# = \prod_{k=1}^n p_k</math>
 
where ''p<sub>k</sub>'' is the ''k''th prime number.
 
For instance, ''p<sub>5</sub>#'' signifies the product of the first 5 primes:
 
:<math>p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310.</math>
 
The first six primorials ''p<sub>n</sub>#'' are:
 
:[[1 (number)|1]], [[2 (number)|2]], [[6 (number)|6]], [[30 (number)|30]], [[210 (number)|210]], 2310. {{OEIS|id=A002110}}
 
The sequence also includes ''p<sub>0</sub>#'' = 1 as [[empty product]].
 
Asymptotically, primorials ''p<sub>n</sub>#'' grow according to:
 
:<math>p_n\# = e^{(1 + o(1)) n \log n},</math>
 
where <math>o(\cdot)</math> is the [[Big O notation#Little-o notation|little-o notation]].<ref name="OEIS A002110" />
 
== Definition for natural numbers ==
In general, for a positive integer ''n'' such a primorial ''n#'' can also be defined, namely as the product of those primes &le; ''n'':<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>
 
:<math>n\# = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>
 
where, <math>\scriptstyle\pi(n)</math> is the [[prime-counting function]] {{OEIS|id=A000720}}, giving the number of primes &le; ''n''.
 
This is equivalent to:
 
:<math>
n\# =
\begin{cases}
    1 & \text{if }n = 1 \\
    n \times ((n-1)\#) & \text{if }n > 1\ \And\ n \text{ is prime} \\
    (n-1)\# & \text{if }n > 1\ \And\ n \text{ is composite}.
\end{cases}
</math>
 
For example, 12# represents the product of those primes &le; 12:
 
:<math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
 
Since <math>\scriptstyle\pi(12)</math> = 5, this can be calculated as:
 
:<math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>
 
Consider the first 12 primorials ''n#'':
 
:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
 
We see that for composite ''n'' every term n# simply duplicates the preceding term (''n''−1)#, as given in the definition. In the above example we have that ''12# &nbsp;=&nbsp; p<sub>5</sub># &nbsp;=&nbsp; 11#'', since 12 is a composite number.
 
Natural logarithm of ''n#'' is the first [[Chebyshev function]], written  <math>\theta(n)</math> or <math>\thetasym(n)</math>, which approaches the linear ''n'' for large ''n''.<ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>
 
Primorials ''n#'' grow according to:
:<math>\ln (n\#) \sim n.</math>
 
The idea of multiplying all known primes occurs in some proofs of the [[infinitude of the prime numbers]], where it is used to derive the existence of another prime.
 
== Applications and properties ==
Primorials play a role in the search for [[Primes in arithmetic progression|prime numbers in additive arithmetic progressions]]. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.
 
Every [[highly composite number]] is a product of primorials (e.g. [[360 (number) | 360]] = 2·6·30).<ref>{{SloanesRef|sequencenumber=A002182|name=Highly composite numbers}}</ref>
 
Primorials are all [[square-free integer]]s, and each one has more distinct [[prime factor]]s than any number smaller than it. For each primorial ''n'', the fraction <math>\phi(n)/n</math> is smaller than for any lesser integer, where <math>\phi</math> is the [[Euler totient function]].
 
Any [[completely multiplicative function]] is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.
 
Primorial base systems (such as base 30) may be useful since they have a lower proportion of [[recurring fraction]]s than any smaller base.
 
== Appearance ==
 
The [[Riemann zeta function]] at positive integers greater than one can be expressed<ref name=mezo/> by using the primorial and the J_k(n) [[Jordan's totient function]]:
: <math> \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)},\quad k=2,3,\dots </math>
 
== Table of primorials ==
 
{| class="wikitable" style="text-align:right"
|-
! ''n''
! ''n''#
! ''p<sub>n</sub>''
! ''p<sub>n</sub>''#
|-
| 0
| 1
| no prime
| 1
|-
| 1
| 1
| 2
| 2
|-
| 2
| 2
| 3
| 6
|-
| 3
| 6
| 5
| 30
|-
| 4
| 6
| 7
| 210
|-
| 5
| 30
| 11
| 2310
|-
| 6
| 30
| 13
| 30030
|-
| 7
| 210
| 17
| 510510
|-
| 8
| 210
| 19
| 9699690
|-
| 9
| 210
| 23
| 223092870
|-
| 10
| 210
| 29
| 6469693230
|-
| 11
| 2310
| 31
| 200560490130
|-
| 12
| 2310
| 37
| 7420738134810
|-
| 13
| 30030
| 41
| 304250263527210
|-
| 14
| 30030
| 43
| 13082761331670030
|-
| 15
| 30030
| 47
| 614889782588491410
|-
| 16
| 30030
| 53
| 32589158477190044730
|-
| 17
| 510510
| 59
| 1922760350154212639070
|-
| 18
| 510510
| 61
| 117288381359406970983270
|-
| 19
| 9699690
| 67
| 7858321551080267055879090
|-
| 20
| 9699690
| 71
| 557940830126698960967415390
|}
 
== See also ==
 
* [[Primorial prime]]
 
== Notes ==
{{reflist|refs=
<ref name=mezo>
{{Cite journal
| last1 = Mező | first1 = István
| title = The Primorial and the Riemann zeta function
| journal = The American Mathematical Monthly
| volume = 120
| issue = 4
| pages = 321
| year = 2013
}}</ref>
}}
 
== References ==
* Harvey Dubner, "Factorial and primorial primes". ''[[Journal of Recreational Mathematics | J. Recr. Math.]]'', 19, 197–203, 1987.
 
[[Category:Integer sequences]]
[[Category:Factorial and binomial topics]]
[[Category:Prime numbers]]
 
[[ru:Факториал#Праймориал или примориал]]

Latest revision as of 16:23, 11 January 2015

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