Primorial
Template:Short description Template:Distinguish
Template:Wikt 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
The primorial <math>p_n\#</math> is defined as the product of the first <math>n</math> primes:<ref name="mathworld">Template:Mathworld</ref><ref name="OEIS A002110">Template:OEIS</ref>
- <math>p_n\# = \prod_{k=1}^n p_k,</math>
where <math>p_k</math> is the <math>k</math>-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:
Asymptotically, primorials grow according to<ref name="OEIS A002110"/>
- <math>p_n\# = e^{(1 + o(1)) n \log n}.</math>
Comparison
The table below shows the comparison between <math>p_n\#</math> and <math>e^{x\log x} = x^x</math>.
| <math>n</math> | <math>p_{n-1}\#</math> | <math>n^n</math> | Absolute error | Relative error |
|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 1 |
| 2 | 2 | 4 | 2 | 2 |
| 3 | 6 | 27 | 21 | 4.5 |
| 4 | 30 | 256 | 226 | 8.53... |
As can be seen, the absolute error and relative error diverges to infinity.
Definition for natural numbers
In general, for a positive integer <math>n</math>, its primorial <math>n\#</math> is the product of all primes less than or equal to <math>n</math>; that is,<ref name="mathworld" /><ref name="OEIS A034386">Template:OEIS</ref>
- <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 Template:OEIS. 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 12:
- <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>n\#</math>:
- <math>1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.</math>
We see that for composite <math>n</math>, every term <math>n\#</math> is equal to the preceding term <math>(n-1)\#</math>. In the above example we have <math>12\# = p_5\# = 11\#</math> since 12 is composite.
Primorials are related to the first Chebyshev function <math>\vartheta(n)</math> by<ref>Template:Mathworld</ref>
- <math>\ln (n\#) = \vartheta(n).</math>
Since <math>\vartheta(n)</math> asymptotically approaches <math>n</math> for large values of <math>n</math>, primorials therefore grow according to:
- <math>n\# = e^{(1+o(1))n}.</math>
Properties
- For any <math>n \in \mathbb{N}</math> such that <math>p\leq n<q</math> for primes <math>p</math> and <math>q</math>, then <math>n\#=p\#</math>.
- Let <math>p_k</math> be the <math>k</math>-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 Template:OEIS.
- 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>.
- <math>\lim_{n \to \infty}\sqrt[n]{n\#} = e </math>. For <math>n<10^{11}</math>, the values are smaller than <math>e</math>,<ref>L. Schoenfeld: Sharper bounds for the Chebyshev functions <math>\theta(x)</math> and <math>\psi(x)</math>. II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef <math>\theta</math> sur le Template:Mvar-ieme nombre premier et grandes valeurs de la fonction <math>\omega(n)</math>, nombre de diviseurs premiers de Template:Mvar. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371</ref> 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 <math>2n</math>, and since <math>\tbinom{2n}{n} \leq 4^{n}</math>, we have the following the upper bound:<ref>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</ref> <math>n\#\leq 4^n</math>.- Using elementary methods, Denis Hanson showed that <math>n\#\leq 3^n</math>.<ref>Template:Cite journal</ref>
- Using more advanced methods, Rosser and Schoenfeld showed that <math>n\#\leq (2.763)^n</math>.<ref name="RosserSchoenfeld1962">Template:Cite journal</ref> Furthermore, they showed that for <math>n \ge 563</math>, <math>n\#\geq (2.22)^n</math>.<ref name="RosserSchoenfeld1962"/>
Applications
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, <math>2 236 133 941+23\#</math> results in a prime, beginning a sequence of thirteen primes found by repeatedly adding <math>23\#</math>, and ending with <math>5136341251</math>. <math>23\#</math> is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials.<ref>Template:Cite OEIS</ref>
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial <math>n</math>, the fraction <math>\varphi(n)/n</math> is smaller than for any positive integer less than <math>n</math>, 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.<ref>Template:Cite journal</ref>
Compositorial
The Template:Mvar-compositorial of a composite number Template:Mvar is the product of all composite numbers up to and including Template:Mvar.<ref name="Wells 2011">Template:Cite book</ref> 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, ...<ref>Template:Cite OEIS</ref>
Riemann zeta function
The Riemann zeta function at positive integers greater than one can be expressed<ref name=mezo/> by using the primorial function and Jordan's totient function <math>J_k</math>:
- <math> \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)},\quad k\in\Z_{>1} </math>.
Table of primorials
See also
Notes
References
- Template:Cite journal
- Spencer, Adam "Top 100" Number 59 part 4.