Linear equations in primes

Abstract

Consider a system of nonconstant affine-linear forms 1 ; : : : ; t W d ! , no two of which are linearly dependent. Let N be a large integer, and let K OE N; N d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N ! 1, for the number of integer points n 2 d \ K for which the integers 1 .n/; : : : ; t .n/ are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime.