primorial (another Prime Pages' Glossary entries) Glossary: Prime Pages: Top 5000: GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)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# = 2.3.5.7.11.13 = 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# = 2.3.5.7 = 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, MultifactorialPrimeRelated pages (outside of this work) Euclid's proof that there are infinitely many primes Chris K. Caldwell © 1999-2019 (all rights reserved)