P. Vishe
DescriptionThis project will focus on learning analytic methods used to solve Diophantine equations. Diophantine equations are polynomial equations with rational coefficients and as a feature of number theory, we will be interested in finding solutions whose co-ordinates are rational numbers. Some of famous questions considered are:1) Waring's problem, which studies whether every large enough integer can be written as a sum of k-th power. N=x_1^k+...+x_n^k.
This can be seen as a
generalisation of Lagrange's four square theorem.
2) Goldbach conjecture: every even number can be written as the sum of two odd primes. Namely, whether N=p_1+p_2,
where we have a linear Diophantine equation but we are looking for more restricted solutions, namely, only taking prime values. A weaker version called as the ternary Goldbach conjecture being that every odd integer can be written as a sum of three odd primes. 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-Ramanujan 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