In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, only prime numbers are multiplied.
There are two conflicting definitions that differ in the interpretation of the argument: the first interprets the argument as an index into the sequence of prime numbers (so that the function is strictly increasing), while the second interprets the argument as a bound on the prime numbers to be multiplied (so that the function value at any composite number is the same as at its predecessor). The rest of this article uses the latter interpretation.
The name "primorial", coined by Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors.
Definition for primorial numbers
where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes:
The first six primorials pn# are:
The sequence also includes p0# = 1 as empty product. Asymptotically, primorials pn# grow according to:
Definition for natural numbers
For example, 12# represents the product of those primes ≤ 12:
Since π(12) = 5, this can be calculated as:
Consider the first 12 primorials n#:
- 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition. In the above example we have 12# = p5# = 11# since 12 is a composite number.
Primorials n# grow according to:
The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, where it is used to derive the existence of another prime.
Applications and properties
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 2136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes. 5
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ(n)/ is smaller than it for any lesser integer, where φ 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.
The n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are
Table of primorials
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- (sequence A002110 in the OEIS)
- (sequence A034386 in the OEIS)
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- Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.
- "Sloane's A036691 : Compositorial numbers: product of first n composite numbers.". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
- Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.