Landau's four problems (1912)

Edmund Landau, ICM Cambridge 1912. Four prime questions he called "unattackable at the present state of mathematics."

Status cutoff 2026-08-17: all four still open. Source: Wikipedia.

#	Problem	Status	What is known
1	Goldbach: every even $n>2$ is $p+q$	Open	Weak (ternary) Goldbach: Helfgott 2013. Chen: large even $n=p+P_2$ . Exceptional set density zero (Montgomery–Vaughan); Pintz $E(x)<x^{0.72}$ .
2	Twin primes: infinitely many $p$ with $p+2$ prime	Open	Zhang 2013 bounded gaps; Polymath to 246. Chen primes: infinitely many $p$ with $p+2$ prime or semiprime.
3	Legendre: a prime in $(n^2,(n+1)^2)$ for every $n$	Open	Equivalent to gaps after $p$ smaller than $2\sqrt{p}$ . Checked through $2^{64}$ via maximal-gap tables. Ingham: a prime between cubes for large $n$ .
4	Infinitely many primes $n^2+1$	Open	OEIS A002496. Iwaniec 1978: infinitely many $n^2+1$ with at most two prime factors. Friedlander–Iwaniec: infinitely many $x^2+y^4$ . Grimmelt–Merikoski 2025: largest prime factor of $n^2+1$ is $\ge n^{1.312}$ i.o. Brun: $O(\sqrt{x}/\log x)$ such primes up to $x$ .

Unrefereed 2026 preprints claiming a proof of (4) are not treated as resolutions.

Hilbert 8 asked for Goldbach and twins as special cases. These four are the smallest classical list that is still entirely open.