Landau’s $n^2+1$ Prime Conjecture

Speaking at the 1912 International Congress of Mathematicians, Landau singled out four problems about primes that he regarded as unattackable by the methods of the day; one of them asks whether $n^2+1$ is prime for infinitely many positive integers $n$ [Landau1912Problems]. It is the simplest case of the general expectation that an irreducible polynomial with no fixed prime divisor should take infinitely many prime values.

Sieve methods come close. Iwaniec proved that $n^2+1$ is a product of at most two primes for infinitely many $n$ [Iwaniec1978Quadratic], and Friedlander and Iwaniec showed that the even sparser polynomial $X^2+Y^4$ represents infinitely many primes [FriedlanderIwaniec1998], demonstrating that thin polynomial sequences are not beyond reach. But the parity barrier prevents classical sieves from separating primes from products of two primes in the one-variable problem.

A proof would seem to need bilinear structure in the sequence $n^2+1$ of the kind exploited for $X^2+Y^4$, or an entirely new idea; the conjecture is open.