## 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*.

| | | | created: 2014-02-20 21:15:29 | modified: 2016-08-28 13:52:44 | by: *bookofproofs*

## 1.Elementary Number Theory

## 2.Analytic Number Theory

## 3.Algebraic Number Theory

## 4.Additive Number Theory

## 5.Some Unsolved Number-Theoretic Problems

## 6.Solving Strategies and Sample Solutions Related to Number Theory

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