Primorial

Template:Short description Template: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

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 Template:Tmathth 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... Template:OEIS.

Asymptotically, primorials grow according to^[2]

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

Definition for natural numbers

In general, for a positive integer Template:Tmath, its primorial $n #$ is the product of all primes less than or equal to Template:Tmath; 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 Template:OEIS. This is equivalent to

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

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

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 Template:Tmath:

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

We see that for composite Template:Tmath, every term $n #$ is equal to the preceding term Template:Tmath. In the above example we have $12 # = p_{5} # = 11 #$ since Template:Tmath 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 Template:Tmath, primorials therefore grow according to:

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

Properties

For any Template:Tmath, $n # = p #$ iff $p$ is the largest prime such that Template:Tmath.
Let $p_{k}$ be the Template:Tmathth prime. Then $p_{k} #$ has exactly $2^{k}$ divisors.
The sum of the reciprocal values of the primorial converges towards a constant
$\sum_{p prime} \frac{1}{p #} = \frac{1}{2} + \frac{1}{6} + \frac{1}{30} + \dots = 0.7052301717918 \dots$ Template:OEIS

The Engel expansion of this number results in the sequence of the prime numbers. Griffiths (2015) proved that it is irrational.^[5]

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 $p$ .
Template:Tmath. For Template:Tmath, the values are smaller than $e$ ,^[6] 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 Template:Tmath, and since (2nn)≤4n, we have the following upper bound:^[7] n#≤4n.
- Using elementary methods, Denis Hanson showed that Template:Tmath.^[8]
- Using more advanced methods, Rosser and Schoenfeld showed that $n # \leq (2.763)^{n}$ .^[9] Furthermore, they showed that for Template:Tmath, Template:Tmath.^[9]

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 Template:Tmath, and ending with Template:Tmath. $23 #$ is also the common difference in arithmetic progressions of fifteen and sixteen primes.

Every highly composite number is a product of primorials.^[10]

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial Template:Tmath, the fraction $φ (n) / n$ is smaller than for any positive integer less than Template:Tmath, 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.^[11]

Compositorial

The Template:Tmath-compositorial of a composite number Template:Tmath is the product of all composite numbers up to and including Template:Tmath.^[12] The Template:Tmath-compositorial is equal to the Template:Tmath-factorial divided by the primorial Template:Tmath. The compositorials are

1, 4, 24, 192, 1728, Template:Val, Template:Val, Template:Val, Template:Val, Template:Val, ...^[13]

Riemann zeta function

The Riemann zeta function at positive integers greater than one can be expressed^[14] by using the primorial function and Jordan's totient function Template:Tmath:

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	Template:Math	Template:Math	Template:Math	Primorial prime?
Template:Mvar	Template:Math	Template:Math	Template:Math	Template:Math^[15]	Template:Math^[16]
0	1	Template:N/a	1	Template:Yes	Template:No
1	1	2	2	Template:Yes	Template:No
2	2	3	6	Template:Yes	Template:Yes
3	6	5	30	Template:Yes	Template:Yes
4	6	7	210	Template:Yes	Template:No
5	30	11	Template:Val	Template:Yes	Template:Yes
6	30	13	Template:Val	Template:No	Template:Yes
7	210	17	Template:Val	Template:No	Template:No
8	210	19	Template:Val	Template:No	Template:No
9	210	23	Template:Val	Template:No	Template:No
10	210	29	Template:Val	Template:No	Template:No
11	Template:Val	31	Template:Val	Template:Yes	Template:No
12	Template:Val	37	Template:Val	Template:No	Template:No
13	Template:Val	41	Template:Val	Template:No	Template:Yes
14	Template:Val	43	Template:Val	Template:No	Template:No
15	Template:Val	47	Template:Val	Template:No	Template:No
16	Template:Val	53	Template:Val	Template:No	Template:No
17	Template:Val	59	Template:Val	Template:No	Template:No
18	Template:Val	61	Template:Val	Template:No	Template:No
19	Template:Val	67	Template:Val	Template:No	Template:No
20	Template:Val	71	Template:Val	Template:No	Template:No
21	Template:Val	73	Template:Val	Template:No	Template:No
22	Template:Val	79	Template:Val	Template:No	Template:No
23	Template:Val	83	Template:Val	Template:No	Template:No
24	Template:Val	89	Template:Val	Template:No	Template:Yes
25	Template:Val	97	Template:Val	Template:No	Template:No
26	Template:Val	101	Template:Val	Template:No	Template:No
27	Template:Val	103	Template:Val	Template:No	Template:No
28	Template:Val	107	Template:Val	Template:No	Template:No
29	Template:Val	109	Template:Val	Template:No	Template:No
30	Template:Val	113	Template:Val	Template:No	Template:No
31	Template:Val	127	Template:Val	Template:No	Template:No
32	Template:Val	131	Template:Val	Template:No	Template:No
33	Template:Val	137	Template:Val	Template:No	Template:No
34	Template:Val	139	Template:Val	Template:No	Template:No
35	Template:Val	149	Template:Val	Template:No	Template:No
36	Template:Val	151	Template:Val	Template:No	Template:No
37	Template:Val	157	Template:Val	Template:No	Template:No
38	Template:Val	163	Template:Val	Template:No	Template:No
39	Template:Val	167	Template:Val	Template:No	Template:No
40	Template:Val	173	Template:Val	Template:No	Template:No

Notes

↑ ^1.0 ^1.1 Template:Mathworld
↑ ^2.0 ^2.1 Template:OEIS
↑ Template:OEIS
↑ Template:Mathworld
↑ Template:Cite journal
↑ 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
↑ Template:Cite journal
↑ ^9.0 ^9.1 Template:Cite journal
↑ Template:Cite OEIS
↑ Template:Cite journal
↑ Template:Cite book
↑ Template:Cite OEIS
↑ Template:Cite journal
↑ Template:Cite OEIS
↑ Template:Cite OEIS

References

Template:Cite journal
Spencer, Adam "Top 100" Number 59 part 4.

Template:Classes of natural numbers Template:Factorial and binomial topics

[mathworld-1] 1.0 ^1.1 Template:Mathworld

[OEIS_A002110-2] 2.0 ^2.1 Template:OEIS

[OEIS_A034386-3] Template:OEIS

[4] Template:Mathworld

[5] Template:Cite journal

[6] 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

[7] 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

[8] Template:Cite journal

[RosserSchoenfeld1962-9] 9.0 ^9.1 Template:Cite journal

[10] Template:Cite OEIS

[11] Template:Cite journal

[Wells_2011-12] Template:Cite book

[13] Template:Cite OEIS

[mezo-14] Template:Cite journal

[15] Template:Cite OEIS

[16] Template:Cite OEIS

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Primorial

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

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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