P. Vishe and A. Mangerel
DescriptionAdditive number theory studies subsets of integers and their behaviour under addition. Two famous, classical problems which are studied are:1) Goldbach conjecture: every even number can be written as the sum of two odd primes. A weaker version called as the ternary Goldbach conjecture being that every odd integer can be written as a sum of three odd primes. 2) Waring's problem, which studies whether every large enough integer can be written as a sum of k-th power. Abstractly, here is a way of looking at it. Let A be an infinite subset of the set of integers Z. One important object to study is the sum-set A+A+...+A;={x_1+...+x_k:x_1,...,x_k
in A}.
We would like to know values taken by the sum sets. The Goldbach conjecture considers the set A= the set of primes and Waring's problem can be seen as a version when A=the set of k-th powers. One of the notable advances of number theory of this century is the resolution of the ternary Goldbach conjecture which was finally finished by Helfgott (2012). The Hardy-Littlewood circle method has been crucial in our understanding of problems of this type. The aim of this project is to give an introduction to the circle method and understanding its applications to additive problems in number theory. Namely, we will roughly have the following objectives: 1) Introduction to the Hardy-Littlewood circle method 2) Application to Waring's problem 3) Proof of ternary Goldbach's conjecture for large enough odd numbers 4) the Hasse principal and solutions to general forms 5) Sieve methods in number theory. Ambitious students can later try to investigate a full proof of the ternary Goldbach conjecture or go deeper into some more recent results in the field. The techniques present a beautiful mix of analysis and number theory. An affinity to both Number theory and Analysis is a must. Prerequisites: Knowledge of Algebra 2, Elementary
Number Theory 2 is necessary. Knowledge of Analysis III
is recommended. Resources: Davenport H. Analytic Methods for Diophantine
Equations and Diophantine Inequalities, (Chapter 3),
Cambridge university press, ISBN: 9780521605830. R. C. Vaughan. The Hardy-Littlewood circle
method.
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email: Pankaj Vishe