Primorial

In mathematics, and more particularly in number theory, primorial, denoted by "${\displaystyle p_{n}\#}$", 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 ${\displaystyle p_{n}\#}$ is defined as the product of the first ${\displaystyle n}$ primes:

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

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

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

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

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

Asymptotically, primorials grow according to

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

Definition for natural numbers

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

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

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

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

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

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

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

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

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

${\displaystyle 1,2,6,6,30,30,210,210,210,210,2310,2310.}$

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

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

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

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

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

Properties

• For any ⁠${\displaystyle n,p\in \mathbb {N} }$⁠, ${\displaystyle n\#=p\#}$ iff ${\displaystyle p}$ is the largest prime such that ⁠${\displaystyle p\leq n}$⁠.
• Let ${\displaystyle p_{k}}$ be the ⁠${\displaystyle k}$⁠th prime. Then ${\displaystyle p_{k}\#}$ has exactly ${\displaystyle 2^{k}}$ divisors.
• The sum of the reciprocal values of the primorial converges towards a constant

  ${\displaystyle \sum _{p\,{\text{prime}}}{1 \over p\#}={1 \over 2}+{1 \over 6}+{1 \over 30}+\ldots =0{.}7052301717918\ldots }$ (sequence A064648 in the OEIS)

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

• Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime ${\displaystyle p}$, the number ${\displaystyle p\#+1}$ has a prime divisor not contained in the set of primes less than or equal to ${\displaystyle p}$.
• ⁠${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n\#}}=e}$⁠. For ⁠${\displaystyle n<10^{11}}$⁠, the values are smaller than ${\displaystyle e}$, but for larger ${\displaystyle n}$, the values of the function exceed ${\displaystyle e}$ and oscillate infinitely around ${\displaystyle e}$ later on.
• Since the binomial coefficient ${\displaystyle {\tbinom {2n}{n}}}$ is divisible by every prime between ${\displaystyle n+1}$ and ⁠${\displaystyle 2n}$⁠, and since ${\displaystyle {\tbinom {2n}{n}}\leq 4^{n}}$, we have the following upper bound: ${\displaystyle n\#\leq 4^{n}}$.
  • Using elementary methods, Denis Hanson showed that ⁠${\displaystyle n\#\leq 3^{n}}$⁠.
  • Using more advanced methods, Rosser and Schoenfeld showed that ${\displaystyle n\#\leq (2.763)^{n}}$. Furthermore, they showed that for ⁠${\displaystyle n\geq 563}$⁠, ⁠${\displaystyle n\#\geq (2.22)^{n}}$⁠.

Applications

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, ${\displaystyle 2236133941+23\#}$ results in a prime, beginning a sequence of thirteen primes found by repeatedly adding ⁠${\displaystyle 23\#}$⁠, and ending with ⁠${\displaystyle 5136341251}$⁠. ${\displaystyle 23\#}$ 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 ⁠${\displaystyle n}$⁠, the fraction ${\displaystyle \varphi (n)/n}$ is smaller than for any positive integer less than ⁠${\displaystyle n}$⁠, where ${\displaystyle \varphi }$ 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 ⁠${\displaystyle n}$⁠-compositorial of a composite number ⁠${\displaystyle n}$⁠ is the product of all composite numbers up to and including ⁠${\displaystyle n}$⁠. The ⁠${\displaystyle n}$⁠-compositorial is equal to the ⁠${\displaystyle n}$⁠-factorial divided by the primorial ⁠${\displaystyle n\#}$⁠. The compositorials are

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

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 ⁠${\displaystyle J_{k}}$⁠:

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

Table of primorials

See also

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

Notes

References

• Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.
• Spencer, Adam "Top 100" Number 59 part 4.