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", attributed to Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors.
Definition for prime 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
This is equivalent to:
For example, 12# represents the product of those primes ≤ 12:
Since , 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 that 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, 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.
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 is smaller than for any lesser integer, 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.
Table of primorials
- Weisstein, Eric W., "Primorial", MathWorld.
- (sequence A002110 in OEIS)
- (sequence A034386 in OEIS)
- Weisstein, Eric W., "Chebyshev Functions", MathWorld.
- "Sloane's A002182 : Highly 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.
- Harvey Dubner, "Factorial and primorial primes". J. Recr. Math., 19, 197–203, 1987.