Suppose the primes are periodic mod $k$, and let $p$ be a prime divisor of $k$. Then there must be infinitely many primes of the form $p + nk$, which is divisible by $p$. Bad news.

EDIT:

More precision: we define $q: \mathbb{Z}\to \mathbb{Z}/k$ to be the canonical quotient and $f: \mathbb{N}\to \mathbb{Z}$ by $f(n) = p_n$, the $n^\mathrm{th}$ prime. Suppose that $g= q\circ f: \mathbb{N}\to \mathbb{Z}/k$ is periodic, with period $t$ and $k> 1$. Let's say $p_m = f(m)$ divides $k$. Then $$ p_{m+t} = f(m+t) \equiv f(m) = p_m\qquad \mathrm{mod}\ k $$ which means $p_{m+t} = p_m + nk$ for some $n$, and this forces $p_m$ to divide $p_{m+t}$, which is impossible. So $g$ cannot be periodic.

This leaves the question: can the primes be eventually periodic?

Suppose the primes are periodic mod $k$, and let $p$ be a prime divisor of $k$. Then there must be infinitely many primes of the form $p + nk$, which is divisible by $p$. Bad news.

EDIT:

More precision: we define $q: \mathbb{Z}\to \mathbb{Z}/k$ to be the canonical quotient and $f: \mathbb{N}\to \mathbb{Z}$ by $f(n) = p_n$, the $n^\mathrm{th}$ prime. Suppose that $g= q\circ f: \mathbb{N}\to \mathbb{Z}/k$ is periodic, with period $t$ and $k> 1$. Let's say $p_m = f(m)$ divides $k$. Then $$ p_{m+t} = f(m+t) \equiv f(m) = p_m\qquad \mathrm{mod}\ k $$ which means $p_{m+t} = p_m + nk$ for some $n$, and this forces $p_m$ to divide $p_{m+t}$, which is impossible. So $g$ cannot be periodic.

This leaves the question: can the primes be eventually periodic?

And of course the dumb answer is yes: mod $2$, the primes are eventually $1,1,1,1,\ldots$.

Suppose the primes are periodic mod $k$, and let $p$ be a prime divisor of $k$. Then there must be infinitely many primes of the form $p + nk$, which is divisible by $p$. Bad news.

Suppose the primes are periodic mod $k$, and let $p$ be a prime divisor of $k$. Then there must be infinitely many primes of the form $p + nk$, which is divisible by $p$. Bad news.

EDIT:

More precision: we define $q: \mathbb{Z}\to \mathbb{Z}/k$ to be the canonical quotient and $f: \mathbb{N}\to \mathbb{Z}$ by $f(n) = p_n$, the $n^\mathrm{th}$ prime. Suppose that $g= q\circ f: \mathbb{N}\to \mathbb{Z}/k$ is periodic, with period $t$ and $k> 1$. Let's say $p_m = f(m)$ divides $k$. Then $$ p_{m+t} = f(m+t) \equiv f(m) = p_m\qquad \mathrm{mod}\ k $$ which means $p_{m+t} = p_m + nk$ for some $n$, and this forces $p_m$ to divide $p_{m+t}$, which is impossible. So $g$ cannot be periodic.

This leaves the question: can the primes be eventually periodic?