Great problems

18th century

1. 1742
   Goldbach conjectureArithmetic and number theory. In a 1742 letter to Euler, Goldbach proposed that every integer greater than 2 is a sum of three primes (counting 1 as prime, as was then customary); Euler reformulated the now-standard, stronger statement that every even integer greater than 2 is a sum of two primes. Despite being checked by computer for every even number up to $4\times10^{18}$, and despite strong partial results, the general statement remains unproved. A separate, weaker statement — the ternary (or weak) Goldbach conjecture, that every odd number greater than 5 is a sum of three primes — was fully proved by Helfgott in 2013.Landau #1Open

19th century

1. 1808
   Legendre's conjectureArithmetic and number theory. Asks whether there is always a prime between any two consecutive squares $n^2$ and $(n+1)^2$ — an interval of length $2n + 1 \approx 2x^{1/2}$ at $x = n^2$. Named by Edmund Landau in 1912 as one of four 'unattackable' prime problems, it remains open even assuming the Riemann hypothesis.Landau #3Open
2. 1849
   Twin prime conjectureArithmetic and number theory. Pairs like $(3,5)$, $(11,13)$, $(17,19)$ are called twin primes. Alphonse de Polignac stated the conjecture in general form in 1849, as the case $n=2$ of the claim that every even $n$ is a gap between infinitely many consecutive primes. The problem was considered hopeless until Yitang Zhang proved in 2013 that some gap below a fixed bound recurs infinitely often — a first, stunning step toward the conjecture. Sieve-theoretic refinements have since brought the bound from 70{,}000{,}000 down to 246, but no method so far can force the gap all the way down to 2.Landau #2Open

20th century

1. 1912
   Primes of the form n² + 1 (Landau's fourth problem)Arithmetic and number theory. Dirichlet's theorem guarantees infinitely many primes in any coprime linear progression $an + b$, yet no single-variable polynomial of degree two or higher has ever been proved to take infinitely many prime values. Posed by Edmund Landau in 1912 as the fourth of his problems on primes, the case $n^2 + 1$ remains open; the closest unconditional result is Iwaniec's 1978 theorem that $n^2 + 1$ is a product of at most two primes infinitely often.Landau #4Open