Let \(f_1, \dots, f_k \in \mathbb{Z}[x]\) be nonconstant irreducible polynomials with positive leading coefficients. Suppose no prime \(p\) divides every value of the product \(F(n) = f_1(n)\cdots f_k(n)\) (no fixed prime divisor). Then there are infinitely many integers \(n\) for which \(f_1(n), \dots, f_k(n)\) are simultaneously prime.
Stated by Andrzej Schinzel and Wacław Sierpiński in 1958, generalising Dirichlet's theorem on primes in arithmetic progressions the way Bunyakovsky's 1857 conjecture generalises it to a single polynomial. Its special cases are a roll-call of famous opens: twin primes \((x, x+2)\), Sophie Germain primes \((x, 2x+1)\), and prime \(k\)-tuples generally; Dickson's conjecture is the linear-polynomial case. The quantitative refinement is the Bateman–Horn conjecture (1962). Not one nonlinear case is known, and even the twin-prime case resists all methods — the Green–Tao theorem finds prime patterns additively, never the exact values Hypothesis H demands.
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