In mathematics, and more particularly in number theory, primorial, denoted by "pn#", 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.
Since asymptotically approaches for large values of , primorials therefore grow according to:
Properties
For any such that for primes and , then .
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 (sequence A064648 in the OEIS).
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 ,[5] 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 the upper bound:[6].
Using elementary methods, mathematician Denis Hanson showed that .[7]
Using more advanced methods, Rosser and Schoenfeld showed that .[8] Furthermore, they showed that for , .[8]
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.
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.
The n-compositorial of a composite numbern is the product of all composite numbers up to and including n.[11] The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are
↑ 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 k-ieme nombre premier et grandes valeurs de la fonction , nombre de diviseurs premiers de n. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p.371
↑ G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. ISBN0-19-853310-1. Theorem 415, p.341
Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math.19: 197–203.
Spencer, Adam "Top 100" Number 59 part 4.
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