Primorial

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Template:Sister project In mathematics, and more particularly in number theory, primorial, denoted by "Template:Math", 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

Template:Math as a function of Template:Math, plotted logarithmically.

For the Template:Mvarth prime number Template:Mvar, the primorial Template:Math is defined as the product of the first Template:Mvar primes:^[1]^[2]

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

,

where Template:Mvar is the Template:Mvarth prime number. For instance, Template:Math signifies the product of the first 5 primes:

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

The first few primorials Template:Math are:

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

Asymptotically, primorials Template:Math grow according to:

p_{n} # = e^{(1 + O (1)) n \log n},

Definition for natural numbers

File:Primorial n plot.png

Template:Math (yellow) as a function of Template:Math, compared to Template:Math(red), both plotted logarithmically.

In general, for a positive integer Template:Mvar, its primorial, Template:Math, is the product of the primes that are not greater than Template:Mvar; that is,^[1]^[3]

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

,

where Template:Math is the prime-counting function (sequence A000720 in the OEIS), which gives the number of primes ≤ Template:Mvar. 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 those primes ≤ 12:

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

Since Template:Math, this can be calculated as:

12 # = p_{π (12)} # = p_{5} # = 2310 .

Consider the first 12 values of Template:Math:

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

We see that for composite Template:Mvar every term Template:Math simply duplicates the preceding term Template:Math, as given in the definition. In the above example we have Template:Math since 12 is a composite number.

Primorials are related to the first Chebyshev function, written Template:Not a typo according to:

\ln (n #) = ϑ (n) .

^[4]

Since Template:Math asymptotically approaches Template:Math for large values of Template:Math, 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 Template:Mvar and Template:Mvar be two adjacent prime numbers. Given any $n \in ℕ$ , where $p \leq n < q$ :

n # = p #

The fact that the binomial coefficient $(\binom{2 n}{n})$ is divisible by every prime between $n + 1$ and $2 n$ , together with the inequality $(\binom{2 n}{n}) \leq 2^{n}$ , allows to derive the upper bound:^[5]

n # \leq 4^{n}

.

Notes:

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

Furthermore:

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

For

n < 1 0^{11}

, the values are smaller than [[e (mathematical constant)|Template:Mvar]],^[8] but for larger Template:Mvar, the values of the function exceed the limit Template:Mvar and oscillate infinitely around Template:Mvar later on.

Let $p_{k}$ be the Template:Mvar-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 ℙ} \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 (See (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 $p$ .

Applications and properties

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, Template:Val + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with Template:Val. 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 Template:Mvar, the fraction Template:Math is smaller than for any lesser integer, where Template:Mvar 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 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 Template:Math. The compositorials are

1, 4, 24, 192, 1728, Template:Val, Template:Val, Template:Val, Template:Val, Template:Val, ...^[12]

Appearance

The Riemann zeta function at positive integers greater than one can be expressed^[13] by using the primorial function and Jordan's totient function Template:Math:

ζ (k) = \frac{2^{k}}{2^{k} - 1} + \sum_{r = 2}^{\infty} \frac{(p_{r - 1} #)^{k}}{J_{k} (p_{r} #)}, k = 2, 3, \dots

Table of primorials

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

Notes

Template:Reflist

References

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Spencer, Adam "Top 100" Number 59 part 4.

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↑ (sequence A034386 in the OEIS)
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↑ 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
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↑ 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
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[OEIS_A002110-2] (sequence A002110 in the OEIS)

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

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[5] 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

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[8] 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

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Primorial

Contents

Definition for prime numbers

Definition for natural numbers

Characteristics

Applications and properties

Appearance

Table of primorials

See also

Notes

References

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Primorial

Definition for prime numbers

Definition for natural numbers

Characteristics

Applications and properties

Appearance

Table of primorials

See also

Notes

References

Navigation menu

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