Primorial: Difference between revisions

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Template:Short description Script error: No such module "Distinguish".

In mathematics, and more particularly in number theory, primorial, denoted by " $p_{n} #$ ", 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

File:Primorial pn plot.png

p n #

Script error: No such module "Check for unknown parameters". as a function of

n

Script error: No such module "Check for unknown parameters"., plotted logarithmically.

The primorial $p_{n} #$ is defined as the product of the first $n$ primes:^[1]^[2]

p_{n} # = \prod_{k = 1}^{n} p_{k},

where $p_{k}$ is the $k$ -th prime number. For instance, $p_{5} #$ signifies the product of the first 5 primes:

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

The first few primorials $p_{n} #$ are:

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... (sequence A002110 in the OEIS).

Asymptotically, primorials grow according to^[2]

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

Definition for natural numbers

File:Primorial n plot.png

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 all primes less than or equal to $n$ ; that is,^[1]^[3]

n # = \prod_{\binom{p \leq n}{p prime}} p = \prod_{i = 1}^{π (n)} p_{i} = p_{π (n)} #,

where $π (n)$ is the prime-counting function (sequence A000720 in the OEIS). This is equivalent to

n # = {\begin{cases} 1 & if n = 0, 1 \\ (n - 1) # \times n & if n is prime \\ (n - 1) # & if n is composite . \end{cases}

For example, $12 #$ represents the product of all primes no greater than 12:

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 .

Consider the first 12 values of the sequence $n #$ :

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

We see that for composite $n$ , every term $n #$ is equal to the preceding term $(n - 1) #$ . In the above example we have $12 # = p_{5} # = 11 #$ since 12 is composite.

Primorials are related to the first Chebyshev function $ϑ (n)$ by^[4]

\ln (n #) = ϑ (n) .

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

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

Properties

For any $n, p \in ℕ$ , $n # = p #$ iff $p$ is the largest prime such that $p \leq n$ .

Let $p_{k}$ be the $k$ -th prime. Then $p_{k} #$ has exactly $2^{k}$ divisors.

The sum of the reciprocal values of the primorial converges towards a constant

\sum_{p prime} \frac{1}{p #} = \frac{1}{2} + \frac{1}{6} + \frac{1}{30} + \dots = 0.7052301717918 \dots

The Engel expansion of this number results in the sequence of the prime numbers (sequence A064648 in the OEIS).

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 $p$ .

$\lim_{n \to \infty} \sqrt[n]{n #} = e$ . For $n < 1 0^{11}$ , the values are smaller than $e$ ,^[5] but for larger $n$ , the values of the function exceed $e$ and oscillate infinitely around $e$ later on.

Since the binomial coefficient (2nn) is divisible by every prime between n+1 and 2n, and since (2nn)≤4n, we have the following upper bound:^[6] n#≤4n.
- Using elementary methods, Denis Hanson showed that $n # \leq 3^{n}$ .^[7]
- Using more advanced methods, Rosser and Schoenfeld showed that $n # \leq (2.763)^{n}$ .^[8] Furthermore, they showed that for $n \geq 563$ , $n # \geq (2.22)^{n}$ .^[8]

Applications

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.^[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 positive integer less than $n$ , 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]

Compositorial

The Template:Mvar-compositorial of a composite number Template:Mvar is the product of all composite numbers up to and including Template:Mvar.^[11] The Template:Mvar-compositorial is equal to the Template:Mvar-factorial divided by the primorial $n #$ Script error: No such module "Check for unknown parameters".. The compositorials are

1, 4, 24, 192, 1728, Script error: No such module "val"., Script error: No such module "val"., Script error: No such module "val"., Script error: No such module "val"., Script error: No such module "val"., ...^[12]

Riemann zeta function

The Riemann zeta function at positive integers greater than one can be expressed^[13] by using the primorial function and Jordan's totient function $J_{k}$ :

ζ (k) = \frac{2^{k}}{2^{k} - 1} + \sum_{r = 2}^{\infty} \frac{(p_{r - 1} #)^{k}}{J_{k} (p_{r} #)}, k \in ℤ_{> 1}

.

Table of primorials

Template:Mvar	$n #$ Script error: No such module "Check for unknown parameters".	Template:Mvar	$p n #$ Script error: No such module "Check for unknown parameters".	Primorial prime?
Template:Mvar		Template:Mvar		p_n# + 1^[14]	p_n# − 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	Script error: No such module "val".	Yes	Yes
6	30	13	Script error: No such module "val".	No	Yes
7	210	17	Script error: No such module "val".	No	No
8	210	19	Script error: No such module "val".	No	No
9	210	23	Script error: No such module "val".	No	No
10	210	29	Script error: No such module "val".	No	No
11	Script error: No such module "val".	31	Script error: No such module "val".	Yes	No
12	Script error: No such module "val".	37	Script error: No such module "val".	No	No
13	Script error: No such module "val".	41	Script error: No such module "val".	No	Yes
14	Script error: No such module "val".	43	Script error: No such module "val".	No	No
15	Script error: No such module "val".	47	Script error: No such module "val".	No	No
16	Script error: No such module "val".	53	Script error: No such module "val".	No	No
17	Script error: No such module "val".	59	Script error: No such module "val".	No	No
18	Script error: No such module "val".	61	Script error: No such module "val".	No	No
19	Script error: No such module "val".	67	Script error: No such module "val".	No	No
20	Script error: No such module "val".	71	Script error: No such module "val".	No	No
21	Script error: No such module "val".	73	Script error: No such module "val".	No	No
22	Script error: No such module "val".	79	Script error: No such module "val".	No	No
23	Script error: No such module "val".	83	Script error: No such module "val".	No	No
24	Script error: No such module "val".	89	Script error: No such module "val".	No	Yes
25	Script error: No such module "val".	97	Script error: No such module "val".	No	No
26	Script error: No such module "val".	101	Script error: No such module "val".	No	No
27	Script error: No such module "val".	103	Script error: No such module "val".	No	No
28	Script error: No such module "val".	107	Script error: No such module "val".	No	No
29	Script error: No such module "val".	109	Script error: No such module "val".	No	No
30	Script error: No such module "val".	113	Script error: No such module "val".	No	No
31	Script error: No such module "val".	127	Script error: No such module "val".	No	No
32	Script error: No such module "val".	131	Script error: No such module "val".	No	No
33	Script error: No such module "val".	137	Script error: No such module "val".	No	No
34	Script error: No such module "val".	139	Script error: No such module "val".	No	No
35	Script error: No such module "val".	149	Script error: No such module "val".	No	No
36	Script error: No such module "val".	151	Script error: No such module "val".	No	No
37	Script error: No such module "val".	157	Script error: No such module "val".	No	No
38	Script error: No such module "val".	163	Script error: No such module "val".	No	No
39	Script error: No such module "val".	167	Script error: No such module "val".	No	No
40	Script error: No such module "val".	173	Script error: No such module "val".	No	No

Notes

↑ ^a ^b Script error: No such module "Template wrapper".
↑ ^a ^b (sequence A002110 in the OEIS)
↑ (sequence A034386 in the OEIS)
↑ Script error: No such module "Template wrapper".
↑ L. Schoenfeld: Sharper bounds for the Chebyshev functions $θ (x)$ and $ψ (x)$ . II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef $θ$ sur le Template:Mvar-ieme nombre premier et grandes valeurs de la fonction $ω (n)$ , nombre de diviseurs premiers de Template:Mvar. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371
↑ G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. Template:ISBN.
Theorem 415, p. 341
↑ Script error: No such module "Citation/CS1".
↑ ^a ^b Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".
↑ Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".

References

Script error: No such module "Citation/CS1".
Spencer, Adam "Top 100" Number 59 part 4.

[mathworld-1] Script error: No such module "Template wrapper".

[OEIS_A002110-2] (sequence A002110 in the OEIS)

[OEIS_A034386-3] (sequence A034386 in the OEIS)

[4] Script error: No such module "Template wrapper".

[5] L. Schoenfeld: Sharper bounds for the Chebyshev functions $θ (x)$ and $ψ (x)$ . II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef $θ$ sur le Template:Mvar-ieme nombre premier et grandes valeurs de la fonction $ω (n)$ , nombre de diviseurs premiers de Template:Mvar. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371

[6] G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. Template:ISBN.
Theorem 415, p. 341

[7] Script error: No such module "Citation/CS1".

[RosserSchoenfeld1962-8] Script error: No such module "Citation/CS1".

[9] Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".

[10] Script error: No such module "Citation/CS1".

[Wells_2011-11] Script error: No such module "citation/CS1".

[12] Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".

[mezo-13] Script error: No such module "Citation/CS1".

[14] Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".

[15] Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".

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@@ Line 2: / Line 2: @@
 {{Distinguish|Primordial (disambiguation){{!}}primordial}}
-{{wikt|-ial}}
+In [[mathematics]], and more particularly in [[number theory]], '''primorial''', denoted by "<math>p_{n}\#</math>", 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.
-In [[mathematics]], and more particularly in [[number theory]], '''primorial''', denoted by "{{math|''p<sub>n</sub>''#}}", 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''.
@@ Line 10: / Line 9: @@
 [[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>
+The primorial <math>p_n\#</math> is defined as the product of the first <math>n</math> 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>,
+:<math>p_n\# = \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:
+where <math>p_k</math> is the <math>k</math>-th prime number. For instance, <math>p_5\#</math> signifies the product of the first 5 primes:
 :<math>p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
-The first few primorials {{math|''p<sub>n</sub>''#}} are:
+The first few primorials <math>p_n\#</math> are:
 :[[1 (number)|1]], [[2 (number)|2]], [[6 (number)|6]], [[30 (number)|30]], [[210 (number)|210]], [[2310 (number)|2310]], 30030, 510510, 9699690... {{OEIS|id=A002110}}.
-Asymptotically, primorials {{math|''p<sub>n</sub>''#}} grow according to:
+Asymptotically, primorials grow according to<ref name="OEIS A002110"/>
-:<math>p_n\# = e^{(1 + \Omicron(1)) n \log n},</math>
+:<math>p_n\# = e^{(1 + o(1)) n \log n}.</math>
 == Definition for natural numbers ==
-[[Image:Primorial n plot.png|thumb|300px|{{math|''n''!}} (yellow) as a function of {{math|''n''}}, compared to {{math|''n''#}}(red), both plotted logarithmically.]]
+[[Image:Primorial n plot.png|thumb|300px|<math>n!</math> (yellow) as a function of <math>n</math>, compared to <math>n\#</math> (red), both plotted logarithmically.]]
-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,<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>
+In general, for a positive integer <math>n</math>, its primorial <math>n\#</math> is the product of all primes less than or equal to <math>n</math>; that is,<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>
-:<math>n\# = \prod_{p \le n\atop p \text{ prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>,
+:<math>n\# = \prod_{p\,\leq\, n\atop p\,\text{prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\#,</math>
-where {{math|''π''(''n'')}} is the [[prime-counting function]] {{OEIS|id=A000720}}, which gives the number of primes ≤ {{mvar|n}}. This is equivalent to:
+where <math>\pi(n)</math> is the [[prime-counting function]] {{OEIS|id=A000720}}. This is equivalent to
 :<math>n\# =
@@ Line 42: / Line 41: @@
 \end{cases}</math>
-For example, 12# represents the product of those primes ≤ 12:
+For example, <math>12\#</math> represents the product of all primes no greater than 12:
 :<math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
-Since {{math|''π''(12) {{=}} 5}}, this can be calculated as:
+Since <math>\pi(12)=5</math>, this can be calculated as:
 :<math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>
-Consider the first 12 values of {{math|''n''#}}:
+Consider the first 12 values of the sequence <math>n\#</math>:
-:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
+:<math>1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.</math>
-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.
+We see that for [[composite number|composite]] <math>n</math>, every term <math>n\#</math> is equal to the preceding term <math>(n-1)\#</math>. In the above example we have <math>12\# = p_5\# = 11\#</math> since 12 is composite.
-Primorials are related to the first [[Chebyshev function]], written {{not a typo|{{math|''{{not a typo|ϑ}}''(''n'')}} or {{math|''θ''(''n'')}}}} according to:
+Primorials are related to the first [[Chebyshev function]] <math>\vartheta(n)</math> by<ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>
-:<math>\ln (n\#) = \vartheta(n).</math><ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>
+:<math>\ln (n\#) = \vartheta(n).</math>
-Since {{math|''{{not a typo|ϑ}}''(''n'')}} asymptotically approaches {{math|''n''}} for large values of {{math|''n''}}, primorials therefore grow according to:
+Since <math>\vartheta(n)</math> asymptotically approaches <math>n</math> for large values of <math>n</math>, primorials therefore grow according to:
-:<math>n\# = e^{(1+\Omicron(1))n}.</math>
+:<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.
+== Properties ==
-== Characteristics ==
+* For any <math>n, p \in \mathbb{N}</math>, <math>n\#=p\#</math> [[iff]] <math>p</math> is the largest prime such that <math>p\leq n</math>.
-* Let {{mvar|p}} and {{mvar|q}} be two adjacent prime numbers. Given any <math>n \in \mathbb{N}</math>, where <math>p\leq n<q</math>:
+* Let <math>p_k</math> be the <math>k</math>-th prime. Then <math>p_k\#</math> has exactly <math>2^k</math> divisors.
-:<math>n\#=p\#</math>
-* The fact that the [[binomial coefficient]] <math>\tbinom{2n}{n}</math> is divisible by every prime between <math>n+1</math> and <math>2n</math>, together with the inequality <math>\tbinom{2n}{n} \leq 2^{n}</math>, allows to derive the upper bound:<ref>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}}.<br />Theorem 415, p.&nbsp;341</ref>
-:<math>n\#\leq 4^n</math>.
-Notes:
+* The sum of the reciprocal values of the primorial [[Convergent series|converges]] towards a constant
+:<math>\sum_{p\,\text{prime}}  {1 \over p\#} = {1 \over 2} + {1 \over 6} + {1 \over 30} + \ldots = 0{.}7052301717918\ldots</math>
+:The [[Engel expansion]] of this number results in the sequence of the prime numbers {{OEIS|A064648}}.
-# Using elementary methods, mathematician Denis Hanson showed that <math>n\#\leq 3^n</math><ref>{{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}}</ref>
+* Euclid's proof of his [[Euclid's theorem|theorem on the infinitude of primes]] can be paraphrased by saying that, for any prime <math>p</math>, the number <math>p\# +1</math> has a prime divisor not contained in the set of primes less than or equal to <math>p</math>.
-# Using more advanced methods, Rosser and Schoenfeld  showed that <math>n\#\leq (2.763)^n</math><ref name="RosserSchoenfeld1962">{{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 }}</ref>
-# Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for <math>n \ge 563</math>, <math>n\#\geq (2.22)^n</math><ref name="RosserSchoenfeld1962"/>
-* Furthermore:
-:<math>\lim_{n \to \infty}\sqrt[n]{n\#} = e </math>
-:For <math>n<10^{11}</math>, the values are smaller than [[e (mathematical constant)|{{mvar|e}}]],<ref>L. Schoenfeld: ''Sharper bounds for the Chebyshev functions <math>\theta(x)</math> and <math>\psi(x)</math>''. II. ''Math. Comp.'' Vol.&nbsp;34, No.&nbsp;134 (1976) 337–360; p.&nbsp;359.<br />Cited in: G. Robin: ''Estimation de la fonction de Tchebychef <math>\theta</math> sur le {{mvar|k}}-ieme nombre premier et grandes valeurs de la fonction <math>\omega(n)</math>, 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.&nbsp;371</ref> but for larger {{mvar|n}}, the values of the function exceed the limit {{mvar|e}} and oscillate infinitely around {{mvar|e}} later on.
-* Let <math>p_k</math> be the {{mvar|k}}-th prime, then <math>p_k\#</math> has exactly <math>2^k</math> divisors. For example, <math>2\#</math> has 2 divisors, <math>3\#</math> has 4 divisors, <math>5\#</math> has 8 divisors and <math>97\#</math> already has <math>2^{25}</math> divisors, as 97 is the 25th prime.
+* <math>\lim_{n \to \infty}\sqrt[n]{n\#} = e </math>. For <math>n<10^{11}</math>, the values are smaller than <math>e</math>,<ref>L. Schoenfeld: ''Sharper bounds for the Chebyshev functions <math>\theta(x)</math> and <math>\psi(x)</math>''. II. ''Math. Comp.'' Vol.&nbsp;34, No.&nbsp;134 (1976) 337–360; p.&nbsp;359.<br />Cited in: G. Robin: ''Estimation de la fonction de Tchebychef <math>\theta</math> sur le {{mvar|k}}-ieme nombre premier et grandes valeurs de la fonction <math>\omega(n)</math>, 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.&nbsp;371</ref> but for larger <math>n</math>, the values of the function exceed <math>e</math> and oscillate infinitely around <math>e</math> later on.
-* The sum of the reciprocal values of the primorial [[Convergent series|converges]] towards a constant
-:<math>\sum_{p\,\in \,\mathbb{P}}  {1 \over p\#} = {1 \over 2} + {1 \over 6} + {1 \over 30} + \ldots = 0{.}7052301717918\ldots</math>
-:The [[Engel expansion]] of this number results in the sequence of the prime numbers (See {{OEIS|A064648}})
-* Euclid's proof of his [[Euclid's theorem|theorem on the infinitude of primes]] can be paraphrased by saying that, for any prime <math>p</math>, the number <math>p\# +1</math> has a prime divisor not contained in the set of primes less than or equal to <math>p</math>.
+* Since the [[binomial coefficient]] <math>\tbinom{2n}{n}</math> is divisible by every prime between <math>n+1</math> and <math>2n</math>, and since <math>\tbinom{2n}{n} \leq 4^{n}</math>, we have the following upper bound:<ref>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}}.<br />Theorem 415, p.&nbsp;341</ref> <math>n\#\leq 4^n</math>.
+**Using elementary methods, Denis Hanson showed that <math>n\#\leq 3^n</math>.<ref>{{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}}</ref>
+** Using more advanced methods, Rosser and Schoenfeld  showed that <math>n\#\leq (2.763)^n</math>.<ref name="RosserSchoenfeld1962">{{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 }}</ref> Furthermore, they showed that for <math>n \ge 563</math>, <math>n\#\geq (2.22)^n</math>.<ref name="RosserSchoenfeld1962"/>
-== Applications and properties ==
+== Applications ==
 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.
+<math>2 236 133 941+23\#</math> results in a prime, beginning a sequence of thirteen primes found by repeatedly adding <math>23\#</math>, and ending with <math>5136341251</math>. <math>23\#</math> 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>
+Every [[highly composite number]] is a product of primorials.<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]].
+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 <math>n</math>, the fraction <math>\varphi(n)/n</math> is smaller than for any positive integer less than <math>n</math>, where <math>\varphi</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.
@@ Line 103: / Line 94: @@
 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>
+== Compositorial ==
 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>
-== Appearance ==
+== Riemann zeta function ==
-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'')}}:
+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_k</math>:
-: <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>
+: <math> \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)},\quad k\in\Z_{>1} </math>.
 == Table of primorials ==
@@ Line 420: / Line 413: @@
 == Notes ==
-{{reflist|refs=
+<references>
 <ref name=mezo>
 {{Cite journal
@@ Line 431: / Line 424: @@
 | year = 2013
 }}</ref>
-}}
+</references>
 == References ==

Primorial: Difference between revisions

Latest revision as of 13:33, 17 December 2025

Contents

Definition for prime numbers

Definition for natural numbers

Properties

Applications

Compositorial

Riemann zeta function

Table of primorials

See also

Notes

References

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Primorial: Difference between revisions

Latest revision as of 13:33, 17 December 2025

Definition for prime numbers

Definition for natural numbers

Properties

Applications

Compositorial

Riemann zeta function

Table of primorials

See also

Notes

References

Navigation menu

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