Project IV (MATH4072) 2026-27

Primes, Goldbach Conjecture and the Circle Method


P. Vishe

Description

Primes are building blocks of numbers and understanding their properties is a key focus of Number Theory. A famous conjecture of Goldbach asks whether every even number can be written as a sum of two primes. This conjecture remains open since 1742 and is one of the most famous classical problems in number theory. A recent breakthrough has established the weaker version of this conjecture, called as the weak Goldbach conjecture. This ambitious project plans to study these problems via studying a key analytic technique called the circle method.

Term 1.

In term 1, the students will study an introduction to the circle method, originated in works of Hardy and Ramanujan (1917) and its application to the weak Goldbach problem. Namely, we will prove the following:

Every large enough odd integer N can be written as a sum of three primes, namely,
N=p_1+p_2+p_3.

Term 2.

In term 2, the student will explore various advanced topics in analytic number theory. Possible choices include:
  • Application of the Circle method to Diophantine Equations, in particular, to Waring's problem.
  • Theory of Exponential sums and applications.
  • Recent advances towards the twin prime conjecture.
  • Hasse Principle and general forms
  • Sieve methods


Mode of Operation and Evidence of Learning.

The project will revolve around learning through reading, examples and problem solving, with focus on mathematical rigour and the development of conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring further examples and applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites: Knowledge of Analysis I, Complex Analysis II, Algebra II is must. Enrollment in Topics in Algebra and Geometry IV (TAG- IV) is recommended. Knowledge of Analysis III may be helpful as well.

Resources:

Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities, (Chapter 3), Cambridge university press, ISBN: 9780521605830.

K. Ram Murty and K. Sinha. Introduction to the circle method, AMS student mathematical library, Volume 104.

email: Pankaj Vishe

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