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pn# as a function of n, plotted logarithmically.
n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied. There are two conflicting definitions that differ in the interpretation of the argument: the first interprets the argument as an index into the sequence of prime numbers (so that the function is strictly increasing), while the second interprets the argument as a bound on the prime numbers to be multiplied (so that the function value at any composite number is the same for its predecessor). The name "primorial" is attributed to Harvey Dubner, and is a portmanteau of prime and factorial.

Contents

Definition for prime numbers

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

p_n\# = \prod_{k=1}^n p_k

where pk is the kth prime number.

For instance, p5# signifies the product of the first 5 primes:

p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310.

The first few primorials pn# are:

1, 2, 6, 30, 210, 2310. (sequence A002110 in OEIS)

The sequence also includes p0# = 1 as empty product.

Asymptotically, primorials pn# grow according to:

p_n\# = \exp \left [ (1 + o(1)) \cdot n \log n \right ],

where "exp" is the exponential function ex and "o" is the little-o notation.[2]

Definition for natural numbers

In general, for a positive integer n such a primorial n# can also be defined, namely as the product of those primes ≤ n:[1][3]

n\# = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\#

where, π(n) is the prime-counting function (sequence A000720 in OEIS), giving the number of primes ≤ n.

This is equivalent to:

 n\# = \begin{cases} 1 & \text{if }n = 1 \ n \times ((n-1)\#) & \text{if }n > 1 \And n \text{ is prime} \ (n-1)\# & \text{if }n > 1 \And n \text{ is composite}. \end{cases}

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

12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.

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

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

Consider the first 12 primorials 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 that 12#  =  p5#  =  11#, since 12 is a composite number.

Natural logarithm of n# is the first Chebyshev function, written θ(n) or \thetasym(n), which approaches the linear n for large n.[4]

Primorials n# grow according to:

\log (n\#) \sim n.

The idea of multiplying all known primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.

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).

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.

Table of primorials

n n# pn pn#
0 1 no prime 1
1 1 2 2
2 2 3 6
3 6 5 30
4 6 7 210
5 30 11 2310
6 30 13 30030
7 210 17 510510
8 210 19 9699690
9 210 23 223092870
10 210 29 6469693230
11 2310 31 200560490130
12 2310 37 7420738134810
13 30030 41 304250263527210
14 30030 43 13082761331670030
15 30030 47 614889782588491410

See also

Notes

  1. ^ a b Weisstein, Eric W., "Primorial" from MathWorld.
  2. ^ a b (sequence A002110 in OEIS)
  3. ^ (sequence A034386 in OEIS)
  4. ^ Weisstein, Eric W., "Chebyshev Functions" from MathWorld.

References

  • Harvey Dubner, "Factorial and primorial primes". J. Recr. Math., 19, 197–203, 1987.
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