Linear equations in primes

Ben Green¹ ; Terence Tao²

¹ Centre for Mathematical Sciences<br/>Wilberforce Road<br/>Cambridge CB3 0WA<br/>United Kingdom
² University of California<br/>Mathematics Department<br/>Los Angeles, CA 90095-1555<br/>United States

Annals of mathematics, Tome 171 (2010) no. 3, pp. 1753-1850.

Résumé

Consider a system $\Psi$ of nonconstant affine-linear forms $\psi_1,\dots,\psi_t: \mathbb{Z}^d \to \mathbb{Z}$, no two of which are linearly dependent. Let $N$ be a large integer, and let $K \subseteq [-N,N]^d$ be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as $N \to \infty$, for the number of integer points $ n \in \mathbb{Z}^d \cap K$ for which the integers $\psi_1( n),\dots,\psi_t( n)$ 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.

Détail

DOI : 10.4007/annals.2010.171.1753