Primorial

In mathematics, and more particularly in number theory, primorial, denoted by "$p_n\mathrm{\#}$", 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# as a function of n, plotted logarithmically.

The primorial $p_n\mathrm{\#}$ is defined as the product of the first $n$ primes:

$p_n\mathrm{\#}=\prod _{k=1}^n{p}_k,$

where $p_k$ is the $k$-th prime number. For instance, $p_5\mathrm{\#}$ signifies the product of the first 5 primes:

$p_5\mathrm{\#}=2\times 3\times 5\times 7\times 11=2310.$

The first few primorials $p_n\mathrm{\#}$ are:

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

Asymptotically, primorials grow according to

$p_n\mathrm{\#}=e^{(1+o(1))n\log n}.$

Definition for natural numbers

File:Primorial n plot.png

$n!$ (yellow) as a function of $n$, compared to $n\mathrm{\#}$ (red), both plotted logarithmically.

In general, for a positive integer $n$, its primorial $n\mathrm{\#}$ is the product of all primes less than or equal to $n$; that is,

$n\mathrm{\#}=\prod _{\genfrac{}{}{0ex}{}{p\le n}{p\text{prime}}}p=\prod _{i=1}^{\pi (n)}{p}_i=p_{\pi (n)}\mathrm{\#},$

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

$n\mathrm{\#}=\{\begin{array}{ll}1& \text{if }n=0,1\\ (n-1)\mathrm{\#}\times n& \text{if }n\text{ is prime}\\ (n-1)\mathrm{\#}& \text{if }n\text{ is composite}.\end{array}$

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

$12\mathrm{\#}=2\times 3\times 5\times 7\times 11=2310.$

Since $\pi (12)=5$, this can be calculated as:

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

Consider the first 12 values of the sequence $n\mathrm{\#}$:

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

We see that for composite $n$, every term $n\mathrm{\#}$ is equal to the preceding term $(n-1)\mathrm{\#}$. In the above example we have $12\mathrm{\#}=p_5\mathrm{\#}=11\mathrm{\#}$ since 12 is composite.

Primorials are related to the first Chebyshev function $\vartheta (n)$ by

$\ln (n\mathrm{\#})=\vartheta (n).$

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

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

Properties

• For any $n,p\in \mathbb{\mathbb{N}}$, $n\mathrm{\#}=p\mathrm{\#}$ iff $p$ is the largest prime such that $p\le n$.

• Let $p_k$ be the $k$-th prime. Then $p_k\mathrm{\#}$ has exactly $2^k$ divisors.

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

$\sum _{p\text{prime}}\frac{1}{p\mathrm{\#}}=\frac{1}{2}+\frac{1}{6}+\frac{1}{30}+\dots =0.7052301717918\dots$

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 $p\mathrm{\#}+1$ has a prime divisor not contained in the set of primes less than or equal to $p$.

• $\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{n\mathrm{\#}}=e$. For $n<10^{11}$, the values are smaller than $e$, but for larger $n$, the values of the function exceed $e$ and oscillate infinitely around $e$ later on.

• Since the binomial coefficient $(\genfrac{}{}{0ex}{}{2n}{n})$ is divisible by every prime between $n+1$ and $2n$, and since $(\genfrac{}{}{0ex}{}{2n}{n})\le 4^n$, we have the following upper bound: $n\mathrm{\#}\le 4^n$.
  • Using elementary methods, Denis Hanson showed that $n\mathrm{\#}\le 3^n$.
  • Using more advanced methods, Rosser and Schoenfeld showed that $n\mathrm{\#}\le (2.763{)}^n$. Furthermore, they showed that for $n\ge 563$, $n\mathrm{\#}\ge (2.22{)}^n$.

Applications

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, $2236133941+23\mathrm{\#}$ results in a prime, beginning a sequence of thirteen primes found by repeatedly adding $23\mathrm{\#}$, and ending with $5136341251$. $23\mathrm{\#}$ is also the common difference in arithmetic progressions of fifteen and sixteen primes.

Every highly composite number is a product of primorials.

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 $\phi (n)/n$ is smaller than for any positive integer less than $n$, where $\phi$ 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.

Compositorial

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

1, 4, 24, 192, 1728, , , , , , ...

Riemann zeta function

The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function $J_k$:

$\zeta (k)=\frac{2^k}{2^k-1}+\sum _{r=2}^{\mathrm{\infty }}\frac{(p_{r-1}\mathrm{\#}{)}^k}{J_k(p_r\mathrm{\#})},k\in {\mathbb{\mathbb{Z}}}_{>1}$.

Table of primorials

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

See also

• Bonse's inequality
• Chebyshev function
• Primorial number system
• Primorial prime

References

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