Parity of Integer equals Parity of Positive Power

Theorem
Let $p \in \Z$ be an integer.

Let $n \in \Z_{>0}$ be a strictly positive integer.

Then $p$ is even iff $p^n$ is even.

That is, the parity of an integer equals the parity of all its (strictly) positive powers.

Proof
Proof by induction:

For all $n \in \Z_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * For all $p \in \Z$, $p$ is even iff $p^n$ is even.

$P \left({1}\right)$ is true, as this just says:
 * $p$ is even iff $p^2$ is even.

Basis for the Induction
$P \left({2}\right)$ is the case:
 * $p$ is even iff $p^2$ is even

which is demonstrated in Parity of Integer equals Parity of its Square.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $p$ is even iff $p^k$ is even

Then we need to show:
 * $p$ is even iff $p^{k+1}$ is even

Induction Step
This is our induction step:

Let $p$ be even.

By the induction hypothesis, $p^k$ is also even.

Then:

and so:

So $p^{k+1}$ is even.

Now suppose $p$ is not even (that is, odd).

By the induction hypothesis, $p^k$ is also not even (that is, odd).

Then:

and so:

So $p^{k+1}$ is not even (that is, odd).

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * For all $n \in \Z_{>0}$, for all $p \in \Z$, $p$ is even iff $p^n$ is even.