Linear Equations in Primes
| dc.creator | Green, Ben | |
| dc.creator | Tao, Terence | |
| dc.date | 2006-06-04 | |
| dc.date | 2008-04-22 | |
| dc.date.accessioned | 2026-07-07T09:33:51Z | |
| dc.date.available | 2026-07-07T09:33:51Z | |
| dc.description | Consider a system Ψof non-constant affine-linear forms ψ_1,...,ψ_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> \infty, for the number of integer points n in K for which the integers ψ_1(n),...,ψ_t(n) are simultaneously prime. This implies many other well-known conjectures, such as the Hardy-Littlewood prime tuples conjecture, the twin prime conjecture, and the (weak) Goldbach conjecture. <p> In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ_1,...,ψ_t are affinely related; this excludes the important ``binary'' cases such as the twin prime or Goldbach conjectures, but does allow one to count ``non-degenerate'' configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the Inverse Gowers-norm conjecture GI(s) and the Mobius and Nilsequences Conjecture MN(s), where s \in {1,2,...} is the complexity of the system and measures the extent to which the forms ψ_i depend on each other. For s = 1 these are essentially classical, and the authors recently resolved the cases s = 2.<p> Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally for any (non-degenerate) problem involving two linear equations in four prime unknowns. | |
| dc.description | 84 pages, numerous small changes made in the light of comments from the referees | |
| dc.identifier | https://arxiv.org/abs/math/0606088 | |
| dc.identifier | http://arxiv.org/abs/math/0606088 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/159283 | |
| dc.subject | Number Theory | |
| dc.subject | Dynamical Systems | |
| dc.title | Linear Equations in Primes | |
| dc.type | text |