In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more particularly in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, primorial, denoted by "#", is a
function from
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s to natural numbers similar to the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...
function, but rather than successively multiplying positive integers, the function only multiplies
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s.
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
For the th prime number , the primorial is defined as the product of the first primes:
:
,
where is the th prime number. For instance, signifies the product of the first 5 primes:
:
The first five primorials are:
:
2,
6,
30,
210
Year 210 ( CCX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Faustinus and Rufinus (or, less frequently, year 963 ''Ab urbe condita ...
,
2310 .
The sequence also includes as
empty product. Asymptotically, primorials grow according to:
:
where is
Little O notation.
Definition for natural numbers
In general, for a positive integer , its primorial, , is the product of the primes that are not greater than ; that is,
:
,
where is the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is ...
, which gives the number of primes ≤ . This is equivalent to:
:
For example, 12# represents the product of those primes ≤ 12:
:
Since , this can be calculated as:
:
Consider the first 12 values of :
:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
We see that for composite every term simply duplicates the preceding term , as given in the definition. In the above example we have since 12 is a composite number.
Primorials are related to the first
Chebyshev function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by
:\vartheta(x)=\sum_ \ln p
where \ln denotes the natural logarithm, ...
, written according to:
:
Since asymptotically approaches for large values of , 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.
Characteristics
* Let and be two adjacent prime numbers. Given any
, where