4th Landau's Problem
Dirichlet's theorem states that
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?