Schinzel's hypothesis H and Bunyakovsky's conjecture

As of 2026, Bunyakovsky's conjecture is unproved for every single polynomial of degree $\ge 2$, and Schinzel's hypothesis H is unproved for every pair of polynomials ($k \ge 2$). The closest unconditional results either relax primality to almost-primality — such as Iwaniec's 1978 theorem that $n^2+1$ has at most two prime factors infinitely often — or replace the single-variable polynomial with a two-variable polynomial of large density, such as the Friedlander–Iwaniec theorem ($a^2+b^4$, 1998) and the Heath-Brown theorem ($a^3+2b^3$, 2001). For linear systems $\{x+h_1, \dots, x+h_k\}$, the Maynard–Tao sieve (2013) proves that for any $m$, if $k$ is large enough, at least $m$ of the linear forms are simultaneously prime infinitely often, establishing a weak existential fragment of Dickson's conjecture.