4th Landau's Problem

Dirichlet's theorem states that

For any two coprime positive integers $a$ and $d$ , there are infinitely many primes of the form $a + nd$ , where $n$ is a non-negative integer.

Argument : There are infinitely many primes of the form $n^2 + 1$ .

Proof : Let $n > 0$ . For $n = 1, n^2 + 1 = 2$ which is prime. Otherwise, if $n^2 + 1 \neq 2$ then $n^2 + 1 = (n)(n) + 1$ and $\gcd(n,1) = 1$ . Thus, by Dirichlet's theorem there are infinitely many prime of the form $(n)(n) + 1$ .

Is this proof of argument (4th Landau's problem) correct?