In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more particularly in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, primorial, denoted by "", is a
function from
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s to natural numbers similar to the
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), 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 ...
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 few primorials are:
:
1,
2,
6,
30,
210,
2310, 30030, 510510, 9699690... .
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 , 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, 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
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work '' Elements''. There are several proofs of the theorem.
Euclid's proof
Euclid of ...
, where it is used to derive the existence of another prime.
Characteristics
* Let and be two adjacent prime numbers. Given any
, where