Dickson's Conjecture

Let \(f_i(n) = a_i n + b_i\) \((i = 1, \dots, k)\) be linear forms with positive integers \(a_i, b_i\), and 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 Leonard Eugene Dickson in 1904 as a sweeping generalisation of Dirichlet's theorem and the twin-prime conjecture. It is the linear-polynomial case of the later Schinzel Hypothesis H (1958): Dickson handles simultaneous primality of linear forms, Schinzel arbitrary polynomials. No nontrivial case is proved — even the twin-prime pair \((x, x+2)\) resists — though the Green–Tao theorem (2004) finds arbitrarily long arithmetic progressions of primes by different means, and Cramér's probabilistic model predicts Dickson-type asymptotics comfortably.