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Primorial#Definition for natural numbers

{{Short description|Product of the first "n" prime numbers}}

{{Distinguish|Primordial (disambiguation){{!}}primordial}}

{{wikt|-ial}}

In mathematics, and more particularly in number theory, primorial, denoted by "{{math|pn#}}", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

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

For the {{mvar|n}}th prime number {{mvar|pn}}, the primorial {{math|pn#}} is defined as the product of the first {{mvar|n}} primes:{{Mathworld | urlname=Primorial | title=Primorial}}{{OEIS|id=A002110}}

:p_n\# = \prod_{k=1}^n p_k,

where {{mvar|pk}} is the {{mvar|k}}th prime number. For instance, {{math|p5#}} signifies the product of the first 5 primes:

:p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.

The first few primorials {{math|pn#}} are:

:1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... {{OEIS|id=A002110}}.

Asymptotically, primorials {{math|pn#}} grow according to:

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

where {{math|o( )}} is Little O notation.

Definition for natural numbers

Image:Primorial n plot.png

In general, for a positive integer {{mvar|n}}, its primorial, {{math|n#}}, is the product of the primes that are not greater than {{mvar|n}}; that is,{{OEIS|id=A034386}}

:n\# = \prod_{p \le n\atop p \text{ prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# ,

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

: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}

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

:12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.

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

:12\# = p_{\pi(12)}\# = p_5\# = 2310.

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# {{=}} p5# {{=}} 11#}} since 12 is a composite number.

Primorials are related to the first Chebyshev function, written {{not a typo|{{math|{{not a typo|ϑ}}(n)}} or {{math|θ(n)}}}} according to:

:\ln (n\#) = \vartheta(n).{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}

Since {{math|{{not a typo|ϑ}}(n)}} asymptotically approaches {{math|n}} for large values of {{math|n}}, primorials therefore grow according to:

:n\# = e^{(1+o(1))n}.

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.

Characteristics

  • Let {{mvar|p}} and {{mvar|q}} be two adjacent prime numbers. Given any n \in \mathbb{N}, where p\leq n:

:n\#=p\#

  • The fact that the binomial coefficient \tbinom{2n}{n} is divisible by every prime between n+1 and 2n, together with the inequality \tbinom{2n}{n} \leq 2^{n}, allows to derive the upper bound:G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. {{ISBN|0-19-853310-1}}.
    Theorem 415, p. 341

:n\#\leq 4^n.

Notes:

  1. Using elementary methods, mathematician Denis Hanson showed that n\#\leq 3^n{{Cite journal |last=Hanson |first=Denis |date=March 1972 |title=On the Product of the Primes |journal=Canadian Mathematical Bulletin |volume=15 |issue=1 |pages=33–37 |doi=10.4153/cmb-1972-007-7|doi-access=free |issn=0008-4395}}
  2. Using more advanced methods, Rosser and Schoenfeld showed that n\#\leq (2.763)^n{{Cite journal |last1=Rosser |first1=J. Barkley |last2=Schoenfeld |first2=Lowell |date=1962-03-01 |title=Approximate formulas for some functions of prime numbers |journal=Illinois Journal of Mathematics |volume=6 |issue=1 |doi=10.1215/ijm/1255631807 |issn=0019-2082|doi-access=free }}
  3. Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for n \ge 563, n\#\geq (2.22)^n
  • Furthermore:

:\lim_{n \to \infty}\sqrt[n]{n\#} = e

:For n<10^{11}, the values are smaller than e (mathematical constant),L. Schoenfeld: Sharper bounds for the Chebyshev functions \theta(x) and \psi(x). II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef \theta sur le {{mvar|k}}-ieme nombre premier et grandes valeurs de la fonction \omega(n), nombre de diviseurs premiers de {{mvar|n}}. Acta Arithm. XLII (1983) 367–389 ([http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4242.pdf PDF 731KB]); p. 371 but for larger {{mvar|n}}, the values of the function exceed the limit {{mvar|e}} and oscillate infinitely around {{mvar|e}} later on.

  • Let p_k be the {{mvar|k}}-th prime, then p_k\# has exactly 2^k divisors. For example, 2\# has 2 divisors, 3\# has 4 divisors, 5\# has 8 divisors and 97\# already has 2^{25} divisors, as 97 is the 25th prime.
  • The sum of the reciprocal values of the primorial converges towards a constant

:\sum_{p\,\in \,\mathbb{P}} {1 \over p\#} = {1 \over 2} + {1 \over 6} + {1 \over 30} + \ldots = 0{.}7052301717918\ldots

:The Engel expansion of this number results in the sequence of the prime numbers (See {{OEIS|A064648}})

  • Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime p, the number p\# +1 has a prime divisor not contained in the set of primes less than or equal to p.

Applications and properties

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance,

{{val|2236133941}} + 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 = {{nowrap|2 × 6 × 30}}).{{Cite OEIS|sequencenumber=A002182|name=Highly composite numbers}}

Primorials are all square-free integers, and each one has more distinct prime factors 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 primorial number system) have a lower proportion of repeating fractions than any smaller base.

Every primorial is a sparsely totient number.{{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 }}

The {{mvar|n}}-compositorial of a composite number {{mvar|n}} is the product of all composite numbers up to and including {{mvar|n}}.{{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}} The {{mvar|n}}-compositorial is equal to the {{mvar|n}}-factorial divided by the primorial {{math|n#}}. The compositorials are

:1, 4, 24, 192, 1728, {{val|17280}}, {{val|207360}}, {{val|2903040}}, {{val|43545600}}, {{val|696729600}}, ...{{Cite OEIS|sequencenumber=A036691|name=Compositorial numbers: product of first n composite numbers.}}

Appearance

The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function {{math|Jk(n)}}:

: \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

Table of primorials

class="wikitable" style="text-align:right"
rowspan="2" | {{mvar|n}}

! rowspan="2" | {{math|n#}}

! rowspan="2" | {{mvar|pn}}

! rowspan="2" | {{math|pn#}}

! colspan="2" | Primorial prime?

pn# + 1{{Cite OEIS|sequencenumber=A014545|name=Primorial plus 1 prime indices}}

! pn# − 1{{Cite OEIS|sequencenumber=A057704|name=Primorial - 1 prime indices}}

0

| 1

| {{n/a}}

| 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

Notes

{{reflist|refs=

{{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

}}

}}

References

  • {{cite journal | last1 = Dubner | first1 = Harvey | year = 1987 | title = Factorial and primorial primes | journal = J. Recr. Math. | volume = 19 | pages = 197–203 }}
  • Spencer, Adam "Top 100" Number 59 part 4.

Category:Integer sequences

Category:Factorial and binomial topics

Category:Prime numbers