Factoring $a^m + 1$, an odd prime

Question

Why is it that if $a^m + 1$, an odd prime, with $m = kl$, and $l$ odd.

We get:

$$a^m + 1 = (a^k + 1)(a^{k(l-1)} - a^{k(l-2)} + \dots + a^k + 1)?$$

What is the name of this property?

Answer 1

If $n$ is odd we have $$\frac{1+x^n}{1+x}=\frac{1-(-x)^n}{1-(-x)}=\frac{1-r^n}{1-r}=\cdots?$$

In your case if $m=kl$ take $x=a^k$. What this is saying is that if $m$ is odd and $a^m+1$ is an odd prime, $m$ must be prime as well.