In mathematics, and more particularly in number theory, primorial, denoted by "<math>p_{n}\#</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
thumb|300px| as a function of , plotted logarithmically.
The primorial <math>p_n\#</math> is defined as the product of the first <math>n</math> primes:
: <math>p_n\# = \prod_{k=1}^n p_k,</math>
where <math>p_k</math> is the 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_n\#</math> are:
: 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... .
Asymptotically, primorials grow according to
: <math>n\# = \prod_{p\,\leq\, n\atop p\,\text{prime p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\#,</math>
where <math>\pi(n)</math> is the prime-counting function . This is equivalent to
: <math>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}</math>
For example, <math>12\#</math> represents the product of all primes no greater than :
: <math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
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 the sequence :
: <math>1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.</math>
We see that for composite , every term <math>n\#</math> is equal to the preceding term . In the above example we have <math>12\# = p_5\# = 11\#</math> since is composite.
Primorials are related to the first Chebyshev function <math>\vartheta(n)</math> by
: <math>\ln (n\#) = \vartheta(n).</math>
Since <math>\vartheta(n)</math> asymptotically approaches <math>n</math> for large values of , primorials therefore grow according to:
: <math>n\# = e^{(1+o(1))n}.</math>
Properties
- For any , <math>n\#=p\#</math> iff <math>p</math> is the largest prime such that .
- Let <math>p_k</math> be the th prime. Then <math>p_k\#</math> has exactly <math>2^k</math> divisors.
- The sum of the reciprocal values of the primorial 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. 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 <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>.
- . For , the values are smaller than <math>e</math>, but for larger <math>n</math>, the values of the function exceed <math>e</math> and oscillate infinitely around <math>e</math> later on.
- Since the binomial coefficient <math>\tbinom{2n}{n}</math> is divisible by every prime between <math>n+1</math> and , and since <math>\tbinom{2n}{n} \leq 4^{n}</math>, we have the following upper bound: <math>n\#\leq 4^n</math>.
- Using elementary methods, Denis Hanson showed that .
- Using more advanced methods, Rosser and Schoenfeld showed that <math>n\#\leq (2.763)^n</math>. Furthermore, they showed that for , .
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial , the fraction <math>\varphi(n)/n</math> is smaller than for any positive integer less than , 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.
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 -compositorial of a composite number is the product of all composite numbers up to and including . The -compositorial is equal to the -factorial divided by the primorial . The compositorials are
:1, 4, 24, 192, 1728, , , , , , ...
Riemann zeta function
The Riemann zeta function at positive integers greater than one can be expressed
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See also
- Bonse's inequality
- Chebyshev function
- Primorial number system
- Primorial prime
Notes
</references>
References
- Spencer, Adam "Top 100" Number 59 part 4.