p_n# as a function of n, plotted logarithmically.

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

For n ≥ 1, the primorial has two similar but distinct meanings. The name is attributed to Harvey Dubner and is a portmanteau of prime and factorial. Harvey Dubner is a semi retired living in New Jersey noted for his contributions to finding large Prime numbers In 1984 he and his son Robert collaborated in developing Definition The factorial function is formally defined by n!=\prod_{k=1}^n k 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)

Asymptotically, primorials p_n# grow according to:

where "exp" is the exponential function e^x and "o" is the "little-o" notation (see Big O notation). Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 In mathematics Six is the second smallest Composite number, its proper Divisors being 1, 2 and 3. 30 ( thirty) is the Natural number following 29 and preceding 31. 210 is the Natural number following 209 and preceding 211. In mathematics 210 is a Composite number, an Abundant The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments ^[2] Its natural logarithm is the first Chebyshev function, written θ(n) or , which approaches the linear n for large n. In Mathematics, the Chebyshev function is either of two related functions ^[3]

In contrast, n# is defined as the product of those primes ≤ n:^[1][4]

This is equivalent to:^[4]

where, π(n) is the prime-counting function (sequence A000720 in OEIS), giving the number of primes ≤ n. In Mathematics, the prime-counting function is the function counting the number of Prime numbers less than or equal to some Real number x The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

For example, 7# represents the product of those primes ≤ 7:

As noted, this can be calculated as:

The first primorials n# are:

1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310

Note that every term n# for composite n simply duplicates the preceding term (n-1)#, as evident in the definition given.

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. Euclid's theorem is a fundamental statement in Number theory which asserts that there are infinitely many Prime numbers There are several well-known proofs

Primorials play a role in the search for prime numbers in additive arithmetic progressions. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members 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. A highly composite number ( HCN) is a positive Integer with more Divisors than any smaller positive integer g. 360 = 2·6·30). 360 ( three hundred and sixty) is the Natural number following 359 and preceding 361

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 In Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder For each primorial n, the fraction φ(n) / n is smaller than for any lesser integer, where φ is the Euler totient function. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to

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. In Number theory, functions of Positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative

Table of primorials

See also

• Primorial prime

References

1. ^ ^{a b} Eric W. Weisstein, Primorial at MathWorld. In Mathematics, primorial primes are Prime numbers of the form pn # ± 1 where pn # is the Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
2. ^ ^{a b} (sequence A002110 in OEIS)
3. ^ Chebyshev Functions - from Wolfram MathWorld
4. ^ ^{a b} (sequence A034386 in OEIS)

• Harvey Dubner, "Factorial and primorial primes". The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences J. Recr. Math., 19, 197–203, 1987. The Journal of Recreational Mathematics is a peer reviewed journal dedicated to Recreational mathematics.