primorial

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, MultifactorialPrime

Related pages (outside of this work)

• Euclid's proof that there are infinitely many primes