Landau’s fourth problem asks whether there exist infinitely many primes of the form 𝑛2 + 1. Despite more than a century of active research, the question remained open. In this paper we give an affirmative answer. The proof is elementary and self-contained. It uses only two classical theorems: Dirichlet’s theorem on primes in arithmetic progressions (1837) and Fermat’s theorem on sums of two squares (1640). No unproven conjectures are required. The key idea is a counting (density) argument. By Dirichlet, there are infinitely many primes 𝑝 ≡ 1 (mod 4). Each such prime can be written uniquely as 𝑝 = 𝑥2 + 𝑦2 with 𝑥 > 𝑦 ≥ 1 (Fermat). If only finitely many of them have 𝑦 = 1 (i.e. 𝑝 = 𝑥2 + 1), then all sufficiently large such primes must satisfy 𝑦 ≥ 2. For any 𝑝 = 𝑥2 + 𝑦2 with 𝑦 ≥ 2 we have 𝑥 ≤ √𝑝; consequently, the number of such primes up to 𝑋 is at most √𝑋. However, the total number of primes 𝑝 ≡ 1 (mod 4) up to 𝑋 is ∼ 𝑋/(2ln𝑋), which for large 𝑋 is much larger than √𝑋. This contradiction shows that our assumption was false: there must be infinitely many primes 𝑝 ≡ 1 (mod 4) with 𝑦 = 1, i.e., infinitely many primes of the form 𝑥2 + 1. Taking 𝑥 even (since 𝑥2 + 1 ≡ 1 (mod 4) forces 𝑥 even) we obtain infinitely many primes 𝑛2 + 1 with 𝑛 even (which is a subset of all primes of this form). The result closes Landau’s fourth problem and adds another classic theorem to the list of solved additive problems for primes.
Andrei Fedotkin (Wed,) studied this question.