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Mathematics Colloquium


Date:  Tuesday, March 07, 2006

Title:  Generalizing the Hardy-Littlewood Method for Primes

Abstract:  The Hardy-Littlewood method, as further developed by Vinogradov, allows one to say rather a lot about a single linear equation in primes. For example one may establish that the number of solutions to p_1 + p_2 + p_3 = N is large if N is a large odd number. One may also show that there are infinitely many non-trivial solutions to the equation p_1 - 2p_2 + p_3 = 0, which describes an arithmetic progression of length 3. I will describe joint work with Terry Tao in which we have been able to generalise this to pairs of linear equations. For example we can count the number of solutions to p_1 - 2p_2 + p_3 = 0, p_2 - 2p_3 + p_4 = 0 (four-term progressions) inside a box. This is part of a programme to handle quite general linear systems in primes, and we will report on the progress towards that. Connections with ergodic theory, combinatorics and (perhaps surprisingly) Lie groups will be alluded to.

Speaker:  Ben Green
Institution:  University of Bristol/Clay Math Institute


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