Primorial

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 numbers

File:Primorial pn plot.png

Template:Math as a function of Template:Math, plotted logarithmically.

For the Template:Mvarth prime number Template:Mvar, the primorial Template:Math is defined as the product of the first Template:Mvar primes:

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

where Template:Mvar is the Template:Mvarth prime number. For instance, Template:Math signifies the product of the first 5 primes:

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

The first five primorials Template:Math are:

2, 6, 30, 210, 2310 Template:OEIS.

The sequence also includes Template:Math as empty product. Asymptotically, primorials Template:Math grow according to:

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

where Template:Math is Little O notation.

Definition for natural numbers

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Template:Math (yellow) as a function of Template:Math, compared to Template:Math(red), both plotted logarithmically.

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,

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

where Template:Math is the prime-counting function Template:OEIS, which gives the number of primes ≤ Template:Mvar. This is equivalent to:

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

For example, 12# represents the product of those primes ≤ 12:

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

Since Template:Math, this can be calculated as:

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

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:

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

Since Template:Math asymptotically approaches Template:Math for large values of Template:Math, primorials therefore grow according to:

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

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.

Characteristics

• Let Template:Mvar and Template:Mvar be two adjacent prime numbers. Given any $n\in \mathbb{N}$, where $p\le n<q$:

$n\mathrm{\#}=p\mathrm{\#}$

• For the Primorial, the following approximation is known:

$n\mathrm{\#}\le 4^n$.

Notes:

1. Using elementary methods, mathematician Denis Hanson showed that $n\mathrm{\#}\le 3^n$
2. Using more advanced methods, Rosser and Schoenfeld showed that $n\mathrm{\#}\le (2.763{)}^n$
3. Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for $n\ge 563$, $n\mathrm{\#}\ge (2.22{)}^n$

• Furthermore:

$\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{n\mathrm{\#}}=e$

For $n<10^{11}$, the values are smaller than [[e (mathematical constant)|Template:Mvar]], but for larger Template:Mvar, the values of the function exceed the limit Template:Mvar and oscillate infinitely around Template:Mvar later on.

• Let $p_k$ be the Template:Mvar-th prime, then $p_k\mathrm{\#}$ has exactly $2^k$ divisors. For example, $2\mathrm{\#}$ has 2 divisors, $3\mathrm{\#}$ has 4 divisors, $5\mathrm{\#}$ has 8 divisors and $97\mathrm{\#}$ already has $2^{25}$ divisors, as 97 is the 25th prime.
• The sum of the reciprocal values of the primorial converges towards a constant

$\sum _{p\in \mathbb{P}}\frac{1}{p\mathrm{\#}}=\frac{1}{2}+\frac{1}{6}+\frac{1}{30}+\dots =0.7052301717918\dots$

The Engel expansion of this number results in the sequence of the prime numbers (See Template:OEIS)

• According to Euclid's theorem, $p\mathrm{\#}+1$ is used to prove the infinitude of the prime numbers.

Applications and properties

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).

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.

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 Template:Math. The compositorials are

1, 4, 24, 192, 1728, Template:Val, Template:Val, Template:Val, Template:Val, Template:Val, ...

Appearance

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

$\zeta (k)=\frac{2^k}{2^k-1}+\sum _{r=2}^{\mathrm{\infty }}\frac{(p_{r-1}\mathrm{\#}{)}^k}{J_k(p_r\mathrm{\#})},k=2,3,\dots$

Table of primorials

Template:Mvar	Template:Math	Template:Mvar	Template:Math	Primorial prime?
				pn# + 1	pn# − 1
0	1	Template:N/a	1	Template:Yes	Template:No
1	1	2	2	Template:Yes	Template:No
2	2	3	6	Template:Yes	Template:Yes
3	6	5	30	Template:Yes	Template:Yes
4	6	7	210	Template:Yes	Template:No
5	30	11	Template:Val	Template:Yes	Template:Yes
6	30	13	Template:Val	Template:No	Template:Yes
7	210	17	Template:Val	Template:No	Template:No
8	210	19	Template:Val	Template:No	Template:No
9	210	23	Template:Val	Template:No	Template:No
10	210	29	Template:Val	Template:No	Template:No
11	Template:Val	31	Template:Val	Template:Yes	Template:No
12	Template:Val	37	Template:Val	Template:No	Template:No
13	Template:Val	41	Template:Val	Template:No	Template:Yes
14	Template:Val	43	Template:Val	Template:No	Template:No
15	Template:Val	47	Template:Val	Template:No	Template:No
16	Template:Val	53	Template:Val	Template:No	Template:No
17	Template:Val	59	Template:Val	Template:No	Template:No
18	Template:Val	61	Template:Val	Template:No	Template:No
19	Template:Val	67	Template:Val	Template:No	Template:No
20	Template:Val	71	Template:Val	Template:No	Template:No
21	Template:Val	73	Template:Val	Template:No	Template:No
22	Template:Val	79	Template:Val	Template:No	Template:No
23	Template:Val	83	Template:Val	Template:No	Template:No
24	Template:Val	89	Template:Val	Template:No	Template:Yes
25	Template:Val	97	Template:Val	Template:No	Template:No
26	Template:Val	101	Template:Val	Template:No	Template:No
27	Template:Val	103	Template:Val	Template:No	Template:No
28	Template:Val	107	Template:Val	Template:No	Template:No
29	Template:Val	109	Template:Val	Template:No	Template:No
30	Template:Val	113	Template:Val	Template:No	Template:No
31	Template:Val	127	Template:Val	Template:No	Template:No
32	Template:Val	131	Template:Val	Template:No	Template:No
33	Template:Val	137	Template:Val	Template:No	Template:No
34	Template:Val	139	Template:Val	Template:No	Template:No
35	Template:Val	149	Template:Val	Template:No	Template:No
36	Template:Val	151	Template:Val	Template:No	Template:No
37	Template:Val	157	Template:Val	Template:No	Template:No
38	Template:Val	163	Template:Val	Template:No	Template:No
39	Template:Val	167	Template:Val	Template:No	Template:No
40	Template:Val	173	Template:Val	Template:No	Template:No

See also

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

References

• Template:Cite journal
• Spencer, Adam "Top 100" Number 59 part 4.