# 4th Landau's Problem

**Number Theory**Level 3

*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?