Let V be an orbit in Zn of a finitely generated subgroup Λ of GLn (Z) whose Zariski closure Zcl (Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the 'congruence graphs' that we associate with V. This expansion property is established when Zcl (Λ) = SL2. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).
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