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The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

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ABSTRACT. In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sketch of the proof of Vinogradov’s three-prime Theorem. 1. ADDITIVE PROBLEMS In the last few centuries many additive problems have come to the attention of mathematicians: famous examples are Waring’s problem and Goldbach’s conjecture. In general, an additive problem can be expressed in the following form: we are given s ≥ 2 subsets of the set of natural numbers N, not necessarily distinct, which we call A1,..., As. We would like to determine the number of solutions of the equation n = a1 + a2 + ·· · + as (1.1) for a given n ∈ N, with the constraint that a j ∈ A j for j = 1,..., s, or, failing that, we would like to prove that the same equation has at least one solution for “sufficiently large ” n. In fact, we can not expect, in general, that for very small n there will be a solution of equation (1.1). Furthermore, depending on the nature of the sets A j, there may be some arithmetical constraints