(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000:
The hardware and software on this system was updated September 4th.  Please let me know of any problem you encounter. <>

We define p# (p-primorial) to be the product of the primes less than or equal to p. For example:

  • 3# = 2.3 = 6,
  • 5# = 2.3.5 = 30, and
  • 13# = = 30030.
This is also called p-prime factorial. Euclid's proof that there are infinitely many primes provides what may be the first use of p# (the concept, not the notation).

It is customary to only apply the notation p# to primes p, but some authors will apply it to any positive real number (e.g., 10.72# = = 210). When viewed this way, the function log(x#) is Tschebycheff's function, and the prime number theorem is equivalent to the expression

log x# ~ x,
(i.e., (log x#)/x approaches 1 as x approaches infinity.)

See Also: Factorial, FactorialPrime, MultifactorialPrime

Related pages (outside of this work)

Chris K. Caldwell © 1999-2014 (all rights reserved)