Template:Short description Template:Distinguish
Template:Wikt 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 numbersEdit
For the Template:Mvarth prime number Template:Mvar, the primorial Template:Math is defined as the product of the first Template:Mvar primes:<ref name="mathworld">Template:Mathworld</ref><ref name="OEIS A002110">(sequence A002110 in the OEIS)</ref>
- <math>p_n\# = \prod_{k=1}^n p_k</math>,
where Template:Mvar is the Template:Mvarth prime number. For instance, Template: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 Template:Math are:
Asymptotically, primorials Template:Math grow according to:
- <math>p_n\# = e^{(1 + o(1)) n \log n},</math>
where Template:Math is Little O notation.<ref name="OEIS A002110" />
Definition for natural numbersEdit
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,<ref name="mathworld" /><ref name="OEIS A034386">(sequence A034386 in the OEIS)</ref>
- <math>n\# = \prod_{p \le n\atop p \text{ prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>,
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:
- <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, 12# represents the product of those primes ≤ 12:
- <math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
Since Template:Math, this can be calculated as:
- <math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>
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:
- <math>\ln (n\#) = \vartheta(n).</math><ref>Template:Mathworld</ref>
Since Template:Math asymptotically approaches Template:Math for large values of Template:Math, primorials therefore grow according to:
- <math>n\# = e^{(1+o(1))n}.</math>
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.
CharacteristicsEdit
- Let Template:Mvar and Template:Mvar be two adjacent prime numbers. Given any <math>n \in \mathbb{N}</math>, where <math>p\leq n<q</math>:
- <math>n\#=p\#</math>
- The fact that the binomial coefficient <math>\tbinom{2n}{n}</math> is divisible by every prime between <math>n+1</math> and <math>2n</math>, together with the inequality <math>\tbinom{2n}{n} \leq 2^{n}</math>, allows to derive 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>.
Notes:
- Using elementary methods, mathematician 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>
- Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for <math>n \ge 563</math>, <math>n\#\geq (2.22)^n</math><ref name="RosserSchoenfeld1962"/>
- Furthermore:
- <math>\lim_{n \to \infty}\sqrt[n]{n\#} = e </math>
- For <math>n<10^{11}</math>, the values are smaller than [[e (mathematical constant)|Template:Mvar]],<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 Template:Mvar, the values of the function exceed the limit Template:Mvar and oscillate infinitely around Template:Mvar later on.
- Let <math>p_k</math> be the Template:Mvar-th prime, then <math>p_k\#</math> has exactly <math>2^k</math> divisors. For example, <math>2\#</math> has 2 divisors, <math>3\#</math> has 4 divisors, <math>5\#</math> has 8 divisors and <math>97\#</math> already has <math>2^{25}</math> divisors, as 97 is the 25th prime.
- The sum of the reciprocal values of the primorial converges towards a constant
- <math>\sum_{p\,\in \,\mathbb{P}} {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 (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 <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>.
Applications and propertiesEdit
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 = Template:Nowrap).<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 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.<ref>Template:Cite journal</ref>
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>
AppearanceEdit
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 Template: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=2,3,\dots </math>
Table of primorialsEdit
See alsoEdit
NotesEdit
ReferencesEdit
- Template:Cite journal
- Spencer, Adam "Top 100" Number 59 part 4.