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{{Distinguish|primordial (disambiguation)}}

In [[mathematics]], and more particularly in [[number theory]], '''primorial''', denoted by “#”, is a [[Function (mathematics)|function]] from [[natural number]]s to natural numbers similar to the [[factorial]] function, but rather than successively multiplying positive integers, the function only multiplies [[prime number]]s. 

The name "primorial", coined by [[Harvey Dubner]], draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''.

== Definition for prime numbers ==
[[Image:Primorial pn plot.png|thumb|300px|{{math|''p<sub>n</sub>''#}} as a function of {{math|''n''}}, plotted logarithmically.]]

For the {{mvar|n}}th prime number {{mvar|p<sub>n</sub>}}, the primorial {{math|''p<sub>n</sub>''#}} is defined as the product of the first {{mvar|n}} primes:<ref name="mathworld">{{Mathworld | urlname=Primorial | title=Primorial}}</ref><ref name="OEIS A002110">{{OEIS|id=A002110}}</ref>

:<math>p_n\# \equiv \prod_{k=1}^n p_k</math>,

where {{mvar|p<sub>k</sub>}} is the {{mvar|k}}th prime number. For instance, {{math|''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 five primorials {{math|''p<sub>n</sub>''#}} are:

:[[2 (number)|2]], [[6 (number)|6]], [[30 (number)|30]], [[210 (number)|210]], 2310 {{OEIS|id=A002110}}.

The sequence also includes {{math|''p''<sub>0</sub># {{=}} 1}} as [[empty product]]. Asymptotically, primorials {{math|''p<sub>n</sub>''#}} grow according to:

:<math>p_n\# = e^{(1 + o(1)) n \log n},</math>

where {{math|''o''( )}} is [[Little O notation]].<ref name="OEIS A002110" />

== Definition for natural numbers ==
[[Image:Primorial n plot.png|thumb|300px|{{math|''n''#}} as a function of {{math|''n''}} (red dots), compared to {{math|''n''!}}, both plotted logarithmically.]]

In general for a positive integer {{mvar|n}}, a primorial {{math|''n''#}} can also be defined, namely as the product of those primes ≤ {{mvar|n}}:<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>

:<math>n\# \equiv \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>,

where {{math|''π''(''n'')}} is the [[prime-counting function]] {{OEIS|id=A000720}}, giving the number of primes ≤ {{mvar|n}}. This is equivalent to:

:<math>n\# = 
\begin{cases}
    1 & \text{if }n = 0,\ 1 \\
    (n-1)\# \times n & \text{if } n \text{ is prime} \\
    (n-1)\# & \text{if } n \text{ is composite}.
\end{cases}</math>

For example, 12# represents the product of those primes ≤ 12:

:<math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>

Since {{math|''π''(12) {{=}} 5}}, this can be calculated as:

:<math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>

Consider the first 12 values of {{math|''n''#}}:

:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.

We see that for composite {{mvar|n}} every term {{math|''n''#}} simply duplicates the preceding term {{math|(''n'' − 1)#}}, as given in the definition. In the above example we have {{math|12# {{=}} ''p''<sub>5</sub># {{=}} 11#}} since 12 is a composite number.

Primorials are related to the first [[Chebyshev function]], written {{math|''ϑ''(''n'')}} or {{math|''θ''(''n'')}} according to:

:<math>\ln (n\#) = \vartheta(n).</math><ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>

Since {{math|''ϑ''(''n'')}} asymptotically approaches {{math|''n''}} for large values of {{math|''n''}}, primorials therefore grow according to:
:<math>n\# = e^{(1+o(1))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, 
{{val|2236133941}}&nbsp;+&nbsp;23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with {{val|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]] = {{nowrap|2 × 6 × 30}}).<ref>{{Cite OEIS|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 {{mvar|n}}, the fraction {{math|{{sfrac|''φ''(''n'')|''n''}}}} is smaller than it for any lesser integer, where {{mvar|φ}} 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.

Base systems corresponding to primorials (such as base 30, not to be confused with the [[Mixed Radix#Primorial number system|primorial number system]]) have a lower proportion of [[repeating fraction]]s than any smaller base.

Every primorial is a [[sparsely totient number]].<ref>{{cite journal | last1=Masser | first1=D.W. | author1-link=David Masser | last2=Shiu | first2=P. | title=On sparsely totient numbers | journal=Pac. J. Math. | volume=121 | pages=407–426 | year=1986 | issn=0030-8730 | zbl=0538.10006 | url=http://projecteuclid.org/euclid.pjm/1102702441 | mr=819198 | doi=10.2140/pjm.1986.121.407}}</ref>

The {{mvar|n}}-compositorial of a [[composite number]] {{mvar|n}} is the product of all composite numbers up to and including {{mvar|n}}.<ref name="Wells 2011">{{cite book|last1=Wells|first1=David|title=Prime Numbers: The Most Mysterious Figures in Math|date=2011|publisher=John Wiley & Sons|isbn=9781118045718|page=29|url=https://books.google.com/books?id=1MTcYrbTdsUC&pg=PA29&lpg=PA29&dq=Compositorial+primorial&source=bl&ots=9G82GeQoX9&sig=SX44vw-N4d3M1s9bVM-W4TyItXI&hl=en&sa=X&ved=0ahUKEwiYtIKE5cXLAhUBU2MKHZLfARkQ6AEIRjAG#v=onepage&q=Compositorial%20primorial&f=false|accessdate=16 March 2016}}</ref> The {{mvar|n}}-compositorial is equal to the {{mvar|n}}-[[factorial]] divided by the primorial {{math|''n''#}}. The compositorials are
:[[1 (number)|1]], [[4 (number)|4]], [[24 (number)|24]], [[192 (number)|192]], 1728, {{val|17280}}, {{val|207360}}, {{val|2903040}}, {{val|43545600}}, {{val|696729600}}, ...<ref>{{Cite OEIS|sequencenumber=A036691|name=Compositorial numbers: product of first n composite numbers.}}</ref>

== Appearance ==

The [[Riemann zeta function]] at positive integers greater than one can be expressed<ref name=mezo/> by using the primorial function and [[Jordan's totient function]] {{math|''J<sub>k</sub>''(''n'')}}:
: <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"
|-
! rowspan="2" | {{mvar|n}}
! rowspan="2" | {{math|''n''#}}
! rowspan="2" | {{mvar|p<sub>n</sub>}}
! rowspan="2" | {{math|''p<sub>n</sub>''#}}<ref>http://planetmath.org/TableOfTheFirst100Primorials</ref>
! colspan="2" | [[Primorial prime]]?
|-
! ''p<sub>n</sub>''# + 1<ref>{{Cite OEIS|sequencenumber=A014545|name=Primorial plus 1 prime indices}}</ref>
! ''p<sub>n</sub>''# − 1<ref>{{Cite OEIS|sequencenumber=A057704|name=Primorial - 1 prime indices}}</ref>
|-
| 0
| 1
| {{n/a}}
| [[Empty product|1]]
| Yes  
| No
|-
| 1
| 1
| 2
| 2
| Yes
| No
|-
| 2
| 2
| 3
| 6
| Yes
| Yes
|-
| 3
| 6
| 5
| 30
| Yes
| Yes
|-
| 4
| 6
| 7
| 210
| Yes
| No
|-
| 5
| 30
| 11
| {{val|2310|fmt=gaps}}
| Yes
| Yes
|-
| 6
| 30
| 13
| {{val|30030}}
| No
| Yes
|-
| 7
| 210
| 17
| {{val|510510}}
| No
| No
|-
| 8
| 210
| 19
| {{val|9699690}}
| No
| No
|-
| 9
| 210
| 23
| {{val|223092870}}
| No
| No
|-
| 10
| 210
| 29
| {{val|6469693230}}
| No
| No
|-
| 11
| {{val|2310|fmt=gaps}}
| 31
| {{val|200560490130}}
| Yes
| No
|-
| 12
| {{val|2310|fmt=gaps}}
| 37
| {{val|7420738134810}}
| No
| No
|-
| 13
| {{val|30030}}
| 41
| {{val|304250263527210}}
| No
| Yes
|-
| 14
| {{val|30030}}
| 43
| {{val|13082761331670030}}
| No
| No
|-
| 15
| {{val|30030}}
| 47
| {{val|614889782588491410}}
| No
| No
|-
| 16
| {{val|30030}}
| 53
| {{val|32589158477190044730}}
| No
| No
|-
| 17
| {{val|510510}}
| 59
| {{val|1922760350154212639070}}
| No
| No
|-
| 18
| {{val|510510}}
| 61
| {{val|117288381359406970983270}}
| No
| No
|-
| 19
| {{val|9699690}}
| 67
| {{val|7858321551080267055879090}}
| No
| No
|-
| 20
| {{val|9699690}}
| 71
| {{val|557940830126698960967415390}}
| No
| No
|-
| 21
| {{val|9699690}}
| 73
| {{val|40729680599249024150621323470}}
| No
| No
|-
| 22
| {{val|9699690}}
| 79
| {{val|3217644767340672907899084554130}}
| No
| No
|-
| 23
| {{val|223092870}}
| 83
| {{val|267064515689275851355624017992790}}
| No
| No
|-
| 24
| {{val|223092870}}
| 89
| {{val|23768741896345550770650537601358310}}
| No
| Yes
|-
| 25
| {{val|223092870}}
| 97
| {{val|2305567963945518424753102147331756070}}
| No
| No
|-
| 26
| {{val|223092870}}
| 101
| {{val|232862364358497360900063316880507363070}}
| No
| No
|-
| 27
| {{val|223092870}}
| 103
| {{val|23984823528925228172706521638692258396210}}
| No
| No
|-
| 28
| {{val|223092870}}
| 107
| {{val|2566376117594999414479597815340071648394470}}
| No
| No
|-
| 29
| {{val|6469693230}}
| 109
| {{val|279734996817854936178276161872067809674997230}}
| No
| No
|-
| 30
| {{val|6469693230}}
| 113
| {{val|31610054640417607788145206291543662493274686990}}
| No
| No
|-
| 31
| {{val|200560490130}}
| 127
| {{val|4014476939333036189094441199026045136645885247730}}
| No
| No
|-
| 32
| {{val|200560490130}}
| 131
| {{val|525896479052627740771371797072411912900610967452630}}
| No
| No
|-
| 33
| {{val|200560490130}}
| 137
| {{val|72047817630210000485677936198920432067383702541010310}}
| No
| No
|-
| 34
| {{val|200560490130}}
| 139
| {{val|10014646650599190067509233131649940057366334653200433090}}
| No
| No
|-
| 35
| {{val|200560490130}}
| 149
| {{val|1492182350939279320058875736615841068547583863326864530410}}
| No
| No
|-
| 36
| {{val|200560490130}}
| 151
| {{val|225319534991831177328890236228992001350685163362356544091910}}
| No
| No
|-
| 37
| {{val|7420738134810}}
| 157
| {{val|35375166993717494840635767087951744212057570647889977422429870}}
| No
| No
|-
| 38
| {{val|7420738134810}}
| 163
| {{val|5766152219975951659023630035336134306565384015606066319856068810}}
| No
| No
|-
| 39
| {{val|7420738134810}}
| 167
| {{val|962947420735983927056946215901134429196419130606213075415963491270}}
| No
| No
|-
| 40
| {{val|7420738134810}}
| 173
| {{val|166589903787325219380851695350896256250980509594874862046961683989710}}
| No
| No
|}

== See also ==
* [[Bonse's inequality]]
* [[Chebyshev function]]
* [[Mixed Radix#Primorial number system|Primorial number system]]
* [[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 ==
* {{cite journal | last1 = Dubner | first1 = Harvey | year = 1987 | title = Factorial and primorial primes | url = | journal = [[Journal of Recreational Mathematics|J. Recr. Math.]] | volume = 19 | issue = | pages = 197–203 }}
*Spencer, Adam “ Top 100 ” Number 59 part 4.

[[Category:Integer sequences]]
[[Category:Factorial and binomial topics]]
[[Category:Prime numbers]]