# 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?

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