Primes of the form n² + 1 (Landau's fourth problem)

As of 2026 Landau's fourth problem remains open: no single-variable polynomial of degree $\ge 2$ has been proved to represent infinitely many primes. The sequence $\{n^2 + 1 : n \le x\}$ has only $\sqrt{x}$ elements up to $x$, which is far sparser than any short interval $[x, x + x^\theta]$ currently accessible, and classical sieve methods are blocked by the parity barrier from distinguishing primes ($P_1$) from products of two primes ($P_2$). The landmark unconditional result remains Henryk Iwaniec's 1978 theorem that $n^2 + 1$ is a $P_2$ almost-prime infinitely often, with $\#\{n \le x : n^2 + 1 \in P_2\} \gg x / \log x$. A second line of attack bounds the largest prime factor $P^+(n^2 + 1)$: if one could reach $P^+(n^2 + 1) > n^2$ infinitely often, $n^2 + 1$ would itself be prime; Jori Merikoski (2023, arXiv:1908.08816) pushed this exponent to $P^+(n^2 + 1) \ge n^{1.279}$ infinitely often (and to $1.312$ in subsequent work). For two-variable polynomials of density $x^{3/4}$ or $x^{2/3}$, the parity barrier has been broken: Friedlander and Iwaniec (1998) proved there are infinitely many primes of the form $a^2 + b^4$, and Heath-Brown (2001) proved the same for $a^3 + 2b^3$.