Two divides Power Plus One iff Odd

Theorem

Let $q, n \in \Z_{>0}$.

Then:

$2 \divides \paren {q^n + 1}$

if and only if $q$ is odd.

In the above, $\divides$ denotes divisibility.

Proof

By Parity of Integer equals Parity of Positive Power, $q^n$ is even if and only if $q$ is even.

Thus it follows that $q^n + 1$ is even if and only if $q$ is odd.

The result follows by definition of even integer.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $8$