Number Theory

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Number Theory

Number Theory is a branch of mathematics dealing with the divisibility properties of integers and in algebraic number fields.

Many problems in number theory can be easily formulated, for instance: What are the integer solutions of a given equation? How many prime numbers are less or equal a given number $n\ge 0$? How many lattice points are there inside a circle/an ellipse? Can every even number be represented as a sum of two prime numbers?

These, and many other number-theoretic questions sound very elementary but turned out to be very hard to answer and many of them have resisted to be answered even until today. However, these hard problems have inspired mathematicians over centuries to develop new ideas and instruments which stimulated even other branches of mathematics.

Theoretical minimum (in a nutshell)

You should be acquainted with arithmetics.
When reading about algebraic number fields, you will have to recap concepts from algebra.
When dealing with analytical number theory, you will have to use some methods from complex analysis and sum manipulation methods.

Concepts you will learn in this part of BookofProofs

In the elementary number theory, we will be dealing with divisibility, prime numbers, characters and methods for solving Diophantine equations.
In the analytical number theory we will be dealing with the distribution of prime numbers and methods to quantify it, including sieve methods and also with the Gamma function and the Riemann hypothesis.
In the algebraic number theory, the concept of divisibility will be extended to general algebraic number fields.
In the additive number theory, we will be dealing with the additive properties of prime numbers and with the progress made in solving the Goldbach hypothesis.