Primorial (original) (raw)

From Wikipedia, the free encyclopedia

Product of the first "n" prime numbers

Look up -ial in Wiktionary, the free dictionary.

In mathematics, and more particularly in number theory, primorial, denoted by "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

[edit]

pn# as a function of n, plotted logarithmically.

For the nth prime number pn, the primorial pn# is defined as the product of the first n primes:[1][2]

p n # = ∏ k = 1 n p k {\displaystyle p_{n}\#=\prod _{k=1}^{n}p_{k}} {\displaystyle p_{n}\#=\prod _{k=1}^{n}p_{k}},

where pk is the kth prime number. For instance, _p_5# signifies the product of the first 5 primes:

p 5 # = 2 × 3 × 5 × 7 × 11 = 2310. {\displaystyle p_{5}\#=2\times 3\times 5\times 7\times 11=2310.} {\displaystyle p_{5}\#=2\times 3\times 5\times 7\times 11=2310.}

The first five primorials pn# are:

2, 6, 30, 210, 2310 (sequence A002110 in the OEIS).

The sequence also includes _p_0# = 1 as empty product. Asymptotically, primorials pn# grow according to:

p n # = e ( 1 + o ( 1 ) ) n log ⁡ n , {\displaystyle p_{n}\#=e^{(1+o(1))n\log n},} {\displaystyle p_{n}\#=e^{(1+o(1))n\log n},}

where o( ) is Little O notation.[2]

Definition for natural numbers

[edit]

n! (yellow) as a function of n, compared to n#(red), both plotted logarithmically.

In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is,[1][3]

n # = ∏ p ≤ n p prime p = ∏ i = 1 π ( n ) p i = p π ( n ) # {\displaystyle n\#=\prod _{p\leq n \atop p{\text{ prime}}}p=\prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#} {\displaystyle n\#=\prod _{p\leq n \atop p{\text{ prime}}}p=\prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#},

where π(n) is the prime-counting function (sequence A000720 in the OEIS), which gives the number of primes ≤ n. This is equivalent to:

n # = { 1 if n = 0 , 1 ( n − 1 ) # × n if n is prime ( n − 1 ) # if n is composite . {\displaystyle 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}}} {\displaystyle 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 × 3 × 5 × 7 × 11 = 2310. {\displaystyle 12\#=2\times 3\times 5\times 7\times 11=2310.} {\displaystyle 12\#=2\times 3\times 5\times 7\times 11=2310.}

Since π(12) = 5, this can be calculated as:

12 # = p π ( 12 ) # = p 5 # = 2310. {\displaystyle 12\#=p_{\pi (12)}\#=p_{5}\#=2310.} {\displaystyle 12\#=p_{\pi (12)}\#=p_{5}\#=2310.}

Consider the first 12 values of 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 12# = _p_5# = 11# since 12 is a composite number.

Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:

ln ⁡ ( n # ) = ϑ ( n ) . {\displaystyle \ln(n\#)=\vartheta (n).} {\displaystyle \ln(n\#)=\vartheta (n).}[4]

Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:

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

n # = p # {\displaystyle n\#=p\#} {\displaystyle n\#=p\#}

n # ≤ 4 n {\displaystyle n\#\leq 4^{n}} {\displaystyle n\#\leq 4^{n}}.

Notes:

  1. Using elementary methods, mathematician Denis Hanson showed that n # ≤ 3 n {\displaystyle n\#\leq 3^{n}} {\displaystyle n\#\leq 3^{n}}[6]
  2. Using more advanced methods, Rosser and Schoenfeld showed that n # ≤ ( 2.763 ) n {\displaystyle n\#\leq (2.763)^{n}} {\displaystyle n\#\leq (2.763)^{n}}[7]
  3. Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for n ≥ 563 {\displaystyle n\geq 563} {\displaystyle n\geq 563}, n # ≥ ( 2.22 ) n {\displaystyle n\#\geq (2.22)^{n}} {\displaystyle n\#\geq (2.22)^{n}}[7]

lim n → ∞ n # n = e {\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n\#}}=e} {\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n\#}}=e}

For n < 10 11 {\displaystyle n<10^{11}} {\displaystyle n<10^{11}}, the values are smaller than e,[8] but for larger n, the values of the function exceed the limit e and oscillate infinitely around e later on.

∑ p ∈ P 1 p # = 1 2 + 1 6 + 1 30 + … = 0 . 7052301717918 … {\displaystyle \sum _{p\,\in \,\mathbb {P} }{1 \over p\#}={1 \over 2}+{1 \over 6}+{1 \over 30}+\ldots =0{.}7052301717918\ldots } {\displaystyle \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 (sequence A064648 in the OEIS))

Applications and properties

[edit]

Primorials play a role in the search for 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 = 2 × 6 × 30).[9]

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction ⁠φ(n)/_n_⁠ is smaller than for any lesser integer, where 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.[10]

The n-compositorial of a composite number n is the product of all composite numbers up to and including n.[11] The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, ...[12]

The Riemann zeta function at positive integers greater than one can be expressed[13] by using the primorial function and Jordan's totient function Jk(n):

ζ ( k ) = 2 k 2 k − 1 + ∑ r = 2 ∞ ( p r − 1 # ) k J k ( p r # ) , k = 2 , 3 , … {\displaystyle \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 } {\displaystyle \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

[edit]

n n# pn pn# Primorial prime?
_pn_# + 1[14] _pn_# − 1[15]
0 1 — 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 2310 Yes Yes
6 30 13 30030 No Yes
7 210 17 510510 No No
8 210 19 9699690 No No
9 210 23 223092870 No No
10 210 29 6469693230 No No
11 2310 31 200560490130 Yes No
12 2310 37 7420738134810 No No
13 30030 41 304250263527210 No Yes
14 30030 43 13082761331670030 No No
15 30030 47 614889782588491410 No No
16 30030 53 32589158477190044730 No No
17 510510 59 1922760350154212639070 No No
18 510510 61 117288381359406970983270 No No
19 9699690 67 7858321551080267055879090 No No
20 9699690 71 557940830126698960967415390 No No
21 9699690 73 40729680599249024150621323470 No No
22 9699690 79 3217644767340672907899084554130 No No
23 223092870 83 267064515689275851355624017992790 No No
24 223092870 89 23768741896345550770650537601358310 No Yes
25 223092870 97 2305567963945518424753102147331756070 No No
26 223092870 101 232862364358497360900063316880507363070 No No
27 223092870 103 23984823528925228172706521638692258396210 No No
28 223092870 107 2566376117594999414479597815340071648394470 No No
29 6469693230 109 279734996817854936178276161872067809674997230 No No
30 6469693230 113 31610054640417607788145206291543662493274686990 No No
31 200560490130 127 4014476939333036189094441199026045136645885247730 No No
32 200560490130 131 525896479052627740771371797072411912900610967452630 No No
33 200560490130 137 72047817630210000485677936198920432067383702541010310 No No
34 200560490130 139 10014646650599190067509233131649940057366334653200433090 No No
35 200560490130 149 1492182350939279320058875736615841068547583863326864530410 No No
36 200560490130 151 225319534991831177328890236228992001350685163362356544091910 No No
37 7420738134810 157 35375166993717494840635767087951744212057570647889977422429870 No No
38 7420738134810 163 5766152219975951659023630035336134306565384015606066319856068810 No No
39 7420738134810 167 962947420735983927056946215901134429196419130606213075415963491270 No No
40 7420738134810 173 166589903787325219380851695350896256250980509594874862046961683989710 No No
  1. ^ a b Weisstein, Eric W. "Primorial". MathWorld.
  2. ^ a b (sequence A002110 in the OEIS)
  3. ^ (sequence A034386 in the OEIS)
  4. ^ Weisstein, Eric W. "Chebyshev Functions". MathWorld.
  5. ^ 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
  6. ^ Hanson, Denis (March 1972). "On the Product of the Primes". Canadian Mathematical Bulletin. 15 (1): 33–37. doi:10.4153/cmb-1972-007-7. ISSN 0008-4395.
  7. ^ a b Rosser, J. Barkley; Schoenfeld, Lowell (1962-03-01). "Approximate formulas for some functions of prime numbers". Illinois Journal of Mathematics. 6 (1). doi:10.1215/ijm/1255631807. ISSN 0019-2082.
  8. ^ L. Schoenfeld: Sharper bounds for the Chebyshev functions θ ( x ) {\displaystyle \theta (x)} {\displaystyle \theta (x)} and ψ ( x ) {\displaystyle \psi (x)} {\displaystyle \psi (x)}. II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
    Cited in: G. Robin: Estimation de la fonction de Tchebychef θ {\displaystyle \theta } {\displaystyle \theta } sur le k-ieme nombre premier et grandes valeurs de la fonction ω ( n ) {\displaystyle \omega (n)} {\displaystyle \omega (n)}, nombre de diviseurs premiers de n. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371
  9. ^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. ^ Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pacific Journal of Mathematics. 121 (2): 407–426. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. MR 0819198. Zbl 0538.10006.
  11. ^ Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A036691 (Compositorial numbers: product of first n composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A014545 (Primorial plus 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A057704 (Primorial - 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.