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## Über dieses Buch

Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed.

Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath.

By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.

## Inhaltsverzeichnis

### Chapter 1. Introduction

In this chapter we review a few different proofs of the Pythagorean Theorem. We also define Pythagorean triples, and explain the types of problems we will be interested in studying in the book.
Ramin Takloo-Bighash

### Chapter 2. Basic number theory

In this chapter we cover basic number theory and set up notations that will be used freely throughout the rest of the book.
Ramin Takloo-Bighash

### Chapter 3. Integral solutions to the Pythagorean Equation

In this chapter we present two different methods to find the solutions of the Pythagorean Equation, one algebraic and one geometric. We then apply the geometric method to find solutions of some other equations.
Ramin Takloo-Bighash

### Chapter 4. What integers are areas of right triangles?

In this chapter we study the set of integers that are the area of a right triangle with integer sides.
Ramin Takloo-Bighash

### Chapter 5. What numbers are the edges of a right triangle?

In this chapter we study numbers that appear as the side lengths of primitive right triangles. We use rings of Gaussian integers to prove our main theorems. We give a quick review of the basic properties of the ring of Gaussian integers. For a more thorough exposition we refer the reader to the classical text by Sierpinski [46] or Conrad [69].
Ramin Takloo-Bighash

### Chapter 6. Primes of the form

The main goal of this chapter is to prove that there are infinitely many primes of the form $$4k+1$$. We will also state the Law of Quadratic Reciprocity.
Ramin Takloo-Bighash

### Chapter 7. Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol

Our first goal in this chapter is to present Gauss’s sixth proof of his Law of Quadratic Reciprocity. The presentation here follows [32, §3.3] fairly closely, except that our Gauss sums are over the complex numbers, as opposed to ibid. where Gauss sums are considered over a finite field. Later in the chapter we introduce the Jacobi symbol and study its basic properties. The Jacobi symbol will make an appearance in Chapter when we give a proof of the Three Squares Theorem.
Ramin Takloo-Bighash

### Chapter 8. Counting Pythagorean triples modulo an integer

In this chapter we consider the Pythagorean Equation in integers modulo a natural number n. In the first section we consider the case where n is a prime number. Later in the chapter we discuss the general case.
Ramin Takloo-Bighash

### Chapter 9. How many lattice points are there on a circle or a sphere?

In this chapter we study the distribution of points with integral coordinates on spheres.
Ramin Takloo-Bighash

### Chapter 10. What about geometry?

In this chapter we present a geometric theorem of Minkowski, and use it to prove Theorem .
Ramin Takloo-Bighash

### Chapter 11. Another proof of the four squares theorem

The goal of this chapter is to give a second proof of the Four Squares Theorem. This proof uses the theory of quaternions, which we will briefly discuss. The proof of the Four Squares Theorem in this chapter is in the spirit of the argument for the Two Squares Theorem we presented in Chapter 5 using Gaussian integers.
Ramin Takloo-Bighash

### Chapter 12. Quadratic forms and sums of squares

Our goal in this chapter is to develop the theory of quadratic forms so we can give another proof of Theorem , especially in the three square case. Our exposition follows [31, Part 3, Chap IV] closely.
Ramin Takloo-Bighash

### Chapter 13. How many Pythagorean triples are there?

In this chapter we determine an asymptotic formula for the number of primitive right triangles with bounded hypotenuse.
Ramin Takloo-Bighash

### Chapter 14. How are rational points distributed, really?

In §3.​2 we found a description of all the points with rational coordinates on the unit circle $$x^2 + y^2 =1$$. In this chapter we examine some topological and analytic properties of these rational points.
Ramin Takloo-Bighash

### Backmatter

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