Primorial

Not to be confused with primordial (disambiguation).

p_n# as a function of n, plotted logarithmically.

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

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 similar to the way the name "factorial" relates to factors.

Definition for primorial numbers

For the nth prime number p_n, the primorial p_n# is defined as the product of the first n primes:

$p_{n}\#\equiv \prod _{k=1}^{n}p_{k}$,

where p_k is the kth prime number. For instance, p₅# signifies the product of the first 5 primes:

$p_{5}\#=2\times 3\times 5\times 7\times 11=2310.$

The first six primorials p_n# are:

1, 2, 6, 30, 210, 2310 (sequence A002110 in the OEIS).

The sequence also includes p₀# = 1 as empty product. Asymptotically, primorials p_n# grow according to:

$p_{n}\#=e^{(1+o(1))n\log n},$

where o( ) is little-o notation.

Definition for natural numbers

In general for a positive integer n, a primorial n# can also be defined, namely as the product of those primes ≤ n:

$n\#\equiv \prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#$,

where π(n) is the prime-counting function (sequence A000720 in the OEIS), giving the number of primes ≤ n. This is equivalent to:

$n\#={\begin{cases}1&{\text{if }}n=0,\ 1\\(n-1)\#\times n&{\text{if }}n{\text{ is prime}}\\(n-1)\#&{\text{if }}n{\text{ is composite}}.\end{cases}}$

For example, 12# represents the product of those primes ≤ 12:

$12\#=2\times 3\times 5\times 7\times 11=2310.$

Since π(12) = 5, this can be calculated as:

$12\#=p_{\pi (12)}\#=p_{5}\#=2310.$

Consider the first 12 values of 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 are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:

$\ln(n\#)=\vartheta (n).$

Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:

$n\#=e^{(1+o(1))n}.$

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, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 7009513634125100000♠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.g. 360 = 2 × 6 × 30).

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)/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.

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.

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

1, 4, 24, 192, 1728, 7004172800000000000♠17280, 7005207360000000000♠207360, 7006290304000000000♠2903040, 7007435456000000000♠43545600, 7008696729600000000♠696729600, ...

Appearance

The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function J_k(n):

$\zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}},\quad k=2,3,\dots$

Table of primorials

n	n#	pn	pn#	Primorial prime?
				pn# + 1	pn# − 1
0	1	N/A	1	Yes	No
1	1	2	2	Yes	No
2	2	3	6	Yes	Yes
3	6	5	30	Yes	Yes
4	6	7	210	Yes	No
5	30	11	7003231000000000000♠2310	Yes	Yes
6	30	13	7004300300000000000♠30030	No	Yes
7	210	17	7005510510000000000♠510510	No	No
8	210	19	7006969969000000000♠9699690	No	No
9	210	23	7008223092870000000♠223092870	No	No
10	210	29	7009646969323000000♠6469693230	No	No
11	7003231000000000000♠2310	31	7011200560490130000♠200560490130	Yes	No
12	7003231000000000000♠2310	37	7012742073813481000♠7420738134810	No	No
13	7004300300000000000♠30030	41	7014304250263527210♠304250263527210	No	Yes
14	7004300300000000000♠30030	43	7016130827613316700♠13082761331670030	No	No
15	7004300300000000000♠30030	47	7017614889782588491♠614889782588491410	No	No
16	7004300300000000000♠30030	53	7019325891584771900♠32589158477190044730	No	No
17	7005510510000000000♠510510	59	7021192276035015421♠1922760350154212639070	No	No
18	7005510510000000000♠510510	61	7023117288381359406♠117288381359406970983270	No	No
19	7006969969000000000♠9699690	67	7024785832155108026♠7858321551080267055879090	No	No
20	7006969969000000000♠9699690	71	7026557940830126698♠557940830126698960967415390	No	No

See also

• Bonse's inequality
• Chebyshev function
• Factorial number system
• Primorial prime

References

• Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.