On the Infinitude of Primes of the Form 𝑛^2 + 1: A Complete Solution to Landau's Fourth Problem

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.