Product of Three Consecutive Integers is never Perfect Power

Theorem
Let $n \in \Z_{> 1}$ be a (strictly) positive integer.

Then:
 * $\left({n - 1}\right) n \left({n + 1}\right)$

cannot be expressed in the form $a^k$ for $a, k \in \Z$ where $k \ge 2$.

That is, the product of $3$ consecutive (strictly) positive integers can never be a perfect power.

Proof
$\left({n - 1}\right) n \left({n + 1}\right) = a^k$ for $a, k \in \Z$ where $k \ge 2$.

We have that:
 * $\gcd \left\{ {(n − 1, n}\right\} = 1 = \gcd \left\{ {(n + 1}\right\}$

Thus $n$ must itself be a perfect power of the form $z^k$ for some $z \in \Z$.

That means $\left({n - 1}\right) \left({n + 1}\right) = n^2 - 1$ must also be a perfect power of the same form.

Let:
 * $n = r^k$ and $n^2 − 1 = s^k$

for $r, s \in \Z$.

Then:
 * $\left({r^2}\right)^k = 1 + s^k$

But the only consecutive integers that are $k$th powers are (trivially) $0$ and $1$.

Hence by Proof by Contradiction there can be no such $n$.