Editing Primorial (section)

== 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 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=Pacific Journal of Mathematics | volume=121 | pages=407–426 | year=1986 | issue=2 | issn=0030-8730 | zbl=0538.10006 | url=http://projecteuclid.org/euclid.pjm/1102702441 | mr=819198 | doi=10.2140/pjm.1986.121.407| doi-access=free }}</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|author-link=David G. Wells|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&q=Compositorial+primorial&pg=PA29|access-date=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 (number)|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>