Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. :

EDIT: As noted in the comments, what I really mean is that the sequence is not pre-periodic; i.e. it is not periodic with some prefix attached. So Jeff Strom's argument doesn't work; and I said that $k>2$ which rules out that case.

Dirichlet's theorem on primes would imply that if it was periodic, then the period would contain each admissible residue class (mod $k$) with equal frequency. For $k=4$, I think there's some result that the quantity $\pi_{1,4}(n) - \pi_{3,4}(n)$ alternates between positive and negative values infinitely often. (Here $\pi_{i,j}(n)$ counts the number of primes less than $n$, congruent to $i$ (mod $j$).) This result settles the question for $k=4$; and perhaps there are generalizations of this to $k>4$. But maybe there's an easier, more elementary approach.

Re Comments: Lucia, thanks for your answers! Your second proof seems elementary (and the result in the first proof is very interesting); but you can include more details? Why does periodicity imply that $\text{log} L(s, \chi)$ is analytic; and why does the corresponding $L$-function have no non-trivial zeros?