Product of Proper Divisors

Theorem
Let $n$ be an integer such that $n \ge 1$.

Let $P \left({n}\right)$ denote the product of the proper divisors of $n$.

Then:
 * $P \left({n}\right) = n^{\tau \left({n}\right) / 2 - 1}$

where $\tau \left({n}\right)$ denotes the $\tau$ function of $n$.

Proof
Let $D \left({n}\right)$ denote the product of all the divisors of $n$.

From Product of Divisors:
 * $D \left({n}\right) = n^{\tau \left({n}\right) / 2}$

The proper divisors of $n$ are defined as being the divisors of $n$ excluding $n$ itself.

Thus:
 * $P \left({n}\right) = \dfrac {D \left({n}\right)} n = \dfrac {n^{\tau \left({n}\right) / 2} } n = n^{\tau \left({n}\right) / 2 - 1}$