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Concept
Primorial
Summary
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.
For the nth prime number pn, the primorial pn# is defined as the product of the first n primes:
where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes:
The first five primorials pn# are:
2, 6, 30, 210, 2310 .
The sequence also includes p0# = 1 as empty product. Asymptotically, primorials pn# grow according to:
where o( ) is Little O notation.
In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is,
where π(n) is the prime-counting function , which gives 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 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 according to:
Since (n) asymptotically approaches n for large values of n, primorials therefore grow according to:
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.
Let p and q be two adjacent prime numbers. Given any , where :
For the Primorial, the following approximation is known:
Notes:
Using elementary methods, mathematician Denis Hanson showed that
Using more advanced methods, Rosser and Schoenfeld showed that
Rosser and Schoenfeld in Theorem 4, formula 3.
Official source
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