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.
Definition for prime numbers
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
Definition for natural numbers
In general, for a positive integer , its primorial is the product of all primes less than or equal to ; that is,
where is the prime-counting function . This is equivalent to
For example, represents the product of all primes no greater than :
Since , this can be calculated as:
Consider the first 12 values of the sequence :
We see that for composite , every term is equal to the preceding term . In the above example we have since is composite.
Primorials are related to the first Chebyshev function by
Since asymptotically approaches for large values of , primorials therefore grow according to:
Properties
For any , iff is the largest prime such that . Let be the th prime. Then has exactly divisors. The sum of the reciprocal values of the primorial converges towards a constant
The Engel expansion of this number results in the sequence of the prime numbers. Griffiths (2015) proved that it is irrational. Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime , the number has a prime divisor not contained in the set of primes less than or equal to . . For , the values are smaller than ,L. Schoenfeld: Sharper bounds for the Chebyshev functions and . II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.Cited in: G. Robin: Estimation de la fonction de Tchebychef sur le -ieme nombre premier et grandes valeurs de la fonction , nombre de diviseurs premiers de . Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371 but for larger , the values of the function exceed and oscillate infinitely around later on. Since the binomial coefficient is divisible by every prime between and , and since , we have the following upper bound:G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. .Theorem 415, p. 341 . Using elementary methods, Denis Hanson showed that . Using more advanced methods, Rosser and Schoenfeld showed that . Furthermore, they showed that for , .
Applications
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, results in a prime, beginning a sequence of thirteen primes found by repeatedly adding , and ending with . is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials.
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial , the fraction is smaller than for any positive integer less than , where is the Euler totient function.
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.
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.
Compositorial
The -compositorial of a composite number is the product of all composite numbers up to and including . The -compositorial is equal to the -factorial divided by the primorial . The compositorials are 1, 4, 24, 192, 1728, , , , , , ...
Riemann zeta function
The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function : .
Table of primorials
Primorial prime? 0 1 1 1 1 2 2 2 2 3 6 3 6 5 30 4 6 7 210 5 30 11 6 30 13 7 210 17 8 210 19 9 210 23 10 210 29 11 31 12 37 13 41 14 43 15 47 16 53 17 59 18 61 19 67 20 71 21 73 22 79 23 83 24 89 25 97 26 101 27 103 28 107 29 109 30 113 31 127 32 131 33 137 34 139 35 149 36 151 37 157 38 163 39 167 40 173
See also Bonse's inequality Chebyshev function Primorial number system Primorial prime
Notes
References
Spencer, Adam "Top 100" Number 59 part 4.