Primorial

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In mathematics, and more particularly in number theory, primorial, denoted by "#", 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

p_n# as a function of n, plotted logarithmically.

For the nth prime number p_n, the primorial p_n# is defined as the product of the first n primes:^[1][2]

[math]\displaystyle{ p_n\# = \prod_{k=1}^n p_k }[/math],

where p_k is the kth prime number. For instance, p₅# signifies the product of the first 5 primes:

[math]\displaystyle{ p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. }[/math]

The first five primorials p_n# are:

2, 6, 30, 210, 2310 (sequence A002110 in the OEIS).

The sequence also includes p₀# = 1 as empty product. Asymptotically, primorials p_n# grow according to:

[math]\displaystyle{ p_n\# = e^{(1 + o(1)) n \log n}, }[/math]

where o( ) is Little O notation.^[2]

Definition for natural numbers

n! (yellow) as a function of n, compared to n#(red), both plotted logarithmically.

In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is,^[1][3]

[math]\displaystyle{ n\# = \prod_{p \le n\atop p \text{ prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# }[/math],

where π(n) is the prime-counting function (sequence A000720 in the OEIS), which gives the number of primes ≤ n. This is equivalent to:

[math]\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} }[/math]

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

[math]\displaystyle{ 12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. }[/math]

Since π(12) = 5, this can be calculated as:

[math]\displaystyle{ 12\# = p_{\pi(12)}\# = p_5\# = 2310. }[/math]

Consider the first 12 values of n#:

1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.

We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition. In the above example we have 12# = p₅# = 11# since 12 is a composite number.

Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:

[math]\displaystyle{ \ln (n\#) = \vartheta(n). }[/math]^[4]

Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:

[math]\displaystyle{ 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.

Characteristics

• Let p and q be two adjacent prime numbers. Given any [math]\displaystyle{ n \in \mathbb{N} }[/math], where [math]\displaystyle{ p\leq n\lt q }[/math]:

[math]\displaystyle{ n\#=p\# }[/math]

• For the Primorial, the following approximation is known:^[5]

[math]\displaystyle{ n\#\leq 4^n }[/math].

Notes:

1. Using elementary methods, mathematician Denis Hanson showed that [math]\displaystyle{ n\#\leq 3^n }[/math]^[6]
2. Using more advanced methods, Rosser and Schoenfeld showed that [math]\displaystyle{ n\#\leq (2.763)^n }[/math]^[7]
3. Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for [math]\displaystyle{ n \ge 563 }[/math], [math]\displaystyle{ n\#\geq (2.22)^n }[/math]^[7]

• Furthermore:

[math]\displaystyle{ \lim_{n \to \infty}\sqrt[n]{n\#} = e }[/math]

For [math]\displaystyle{ n\lt 10^{11} }[/math], the values are smaller than e,^[8] but for larger n, the values of the function exceed the limit e and oscillate infinitely around e later on.

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

[math]\displaystyle{ \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))

• According to Euclid's theorem, [math]\displaystyle{ p\# +1 }[/math] 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, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 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 = 2 × 6 × 30).^[9]

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ(n)/n is smaller than for any lesser integer, where φ 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.^[10]

The n-compositorial of a composite number n is the product of all composite numbers up to and including n.^[11] The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, ...^[12]

Appearance

The Riemann zeta function at positive integers greater than one can be expressed^[13] by using the primorial function and Jordan's totient function J_k(n):

[math]\displaystyle{ \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 primorials

See also

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

Notes

References

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

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