p_n# as a function of n, plotted logarithmically.

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

The name primorial is attributed to Harvey Dubner and is a portmanteau of prime and factorial. There are two definitions for primorial: one for prime numbers and one for natural numbers.

[edit] Definition for prime numbers

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

where p_k is the kth prime number.

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

The first few primorials p_n# are:

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

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

Asymptotically, primorials p_n# grow according to:

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

[edit] 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]

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

This is equivalent to:

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

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

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 , which approaches the linear n for large n.^[4]

Primorials n# grow according to:

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.

[edit] 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.

[edit] Table of primorials

[edit] See also

• Primorial prime

[edit] 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.

[edit] References

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