primorial

English

Comparison of factorial (yellow) and primorial, both plotted logarithmically

Etymology

Blend of prime +‎ factorial. Coined by American engineer and mathematician Harvey Dubner.

Noun

primorial (plural primorials)

1. (number theory) Any number belonging to the integer sequence whose nth element is the product of the first n primes.

   Synonym: primorial number

   • 2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 97:

     For example, up to $10^{9}$, every positive integer which is a multiple of a primorial and is less than the following primorial satisfies the given inequality. There has also been little discussion on which primorials qualify in the different situations when RH fails.

2. (number theory) A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input.

   The primorial of $6$ is $\textstyle 6\#=2\times 3\times 5=30$.

   • 2020, Rong Pan, Qinheping Hu, Rishabh Singh, Loris D'Antoni, Solving Problem Sketches with Large Integer Values, Peter Müller (editor), Programming Languages and Systems: 29th European Symposium, Proceedings, Springer, LNCS 12075, page 587,

     The following number theory result relates the primorial to the Chebyshev function.

     $\vartheta (n)=\log(n\#)=\log {2^{(1+o(n))n}}=(1+o(n))n$

Usage notes

The primorial operation may be defined as:

$n\#=\prod _{p\leq n \atop p~{\mathsf {prime}}}p=\prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#$,

where $\pi (n)$ denotes the prime-counting function, which gives the number of primes $\leq n$.

It can also be defined recursively:

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

Related terms

• primorial prime

Translations

product of primes

• French: please add this translation if you can
• German: Primorial f, Primfakultät f (rare)
• Italian: primoriale (it) m
• Turkish: primoriyel

Further reading

• Sequence A002110 of the On-Line Encyclopedia of Integer Sequences
• Chebyshev function on Wikipedia.Wikipedia (Particularly, first Chebyshev function.)
• Sparsely totient number on Wikipedia.Wikipedia