TITLE: Choosing $a$ s.t. $\frac{a^k - 1}{a-1}$ is not a prime power
QUESTION [5 upvotes]: Let us suppose that we are presented with a positive integer $k$ and asked to come up with a positive integer $a$ such that $\frac{a^k - 1}{a-1}$ is not a prime power, or just to prove in an elementary way that this can be done. How do we proceed?

Of course, in practice this is essentially trivial. But is there a cannonical choice? For that matter, is there an elementary proof that shows that this is always possible? 
This came up in a discussion with another Brown mathie about interesting things to talk about in an elementary number theory class I will be teaching this summer. To be more specific, we were trying to prove the infinitude of primes in arithmetic progressions of the form $1, 1+p^n, 1+2p^n, ...$ using a naive a completely elementary approach (hopefully to be replicated by my students this summer), and this side topic came up.

REPLY [2 votes]: Let $a=2k+1$ then
$$
\frac{a^k-1}{a-1} = \frac{\sum_{i=1}^k\binom{k}{i}(2k)^i}{2k}\equiv k\pmod{k^2}
$$
which is not $k$, divisible by $k$ but not $k^2$, and hence not a prime or a power for $k>1$.