Primorial
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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: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: : 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, 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, :n\# = \prod_ p = \prod_^ p_i = p_\# , where is the prime-counting function , which gives the number of primes ≤ . This is equivalent to: :n\# = \begin 1 & \textn = 0,\ 1 \\ (n-1)\# \times n & \text n \text \\ (n-1)\# & \text n \text. \end For example, 12# represents the product of those primes ≤ 12: :12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. Since , this can be calculated as: :12\# = p_\# = p_5\# = 2310. 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: :\ln (n\#) = \vartheta(n). Since asymptotically approaches for large values of , primorials therefore grow according to: :n\# = e^. 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 n \in \mathbb, where p\leq n: :n\#=p\# * The fact that the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
\tbinom is divisible by every prime between n+1 and 2n, together with the inequality \tbinom \leq 2^, allows to derive the upper bound: :n\#\leq 4^n. Notes: # Using elementary methods, mathematician Denis Hanson showed that n\#\leq 3^n # Using more advanced methods, Rosser and Schoenfeld showed that n\#\leq (2.763)^n # Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for n \ge 563, n\#\geq (2.22)^n * Furthermore: :\lim_\sqrt = e :For n<10^, the values are smaller than , but for larger , the values of the function exceed the limit and oscillate infinitely around later on. * Let p_k be the -th prime, then p_k\# has exactly 2^k divisors. For example, 2\# has 2 divisors, 3\# has 4 divisors, 5\# has 8 divisors and 97\# already has 2^ divisors, as 97 is the 25th prime. * The sum of the reciprocal values of the primorial converges towards a constant :\sum_ = + + + \ldots = 07052301717918\ldots :The
Engel expansion The Engel expansion of a positive real number ''x'' is the unique non-decreasing sequence of positive integers (a_1,a_2,a_3,\dots) such that :x=\frac+\frac+\frac+\cdots = \frac\!\left(1 + \frac\!\left(1 + \frac\left(1+\cdots\right)\right)\right) ...
of this number results in the sequence of the prime numbers (See ) * Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime p, the number p\# +1 has a prime divisor not contained in the set of primes less than or equal to p.


Applications and properties

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance,  + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with . 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.
360 360 may refer to: * 360 (number) * 360 AD, a year * 360 BC, a year * 360 degrees, a turn Businesses and organizations * 360 Architecture, an American architectural design firm * Ngong Ping 360, a tourism project in Lantau Island, Hong Kong ...
= ). Primorials are all square-free integers, and each one has more distinct
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
s than any number smaller than it. For each primorial , the fraction is smaller than for any lesser integer, where is the Euler totient function. Any
completely multiplicative function In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
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. Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system) have a lower proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number. The -compositorial of a
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
is the product of all composite numbers up to and including . The -compositorial is equal 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 ...
divided by the primorial . The compositorials are : 1, 4, 24, 192, 1728, , , , , , ...


Appearance

The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function : : \zeta(k)=\frac+\sum_^\infty\frac,\quad k=2,3,\dots


Table of primorials


See also

* Bonse's inequality * Chebyshev function * Primorial number system * Primorial prime


Notes


References

* {{cite journal , last1 = Dubner , first1 = Harvey , year = 1987 , title = Factorial and primorial primes , journal = J. Recr. Math. , volume = 19 , pages = 197–203 *Spencer, Adam "Top 100" Number 59 part 4. Integer sequences Factorial and binomial topics Prime numbers