Right Triangles, Sums of Squares, and Arithmetic

Preface
Contents
Notation
Part I Foundational material
1 Introduction
	1.1 The Pythagorean Theorem
	1.2 Pythagorean triples
	1.3 The questions
2 Basic number theory
	2.1 Natural numbers, mathematical induction,  and the Well-ordering Principle
	2.2 Divisibility and prime factorization
	2.3 The Chinese Remainder Theorem
	2.4 Euler\'s Theorem
	2.5 Polynomials modulo a prime
	2.6 Digit expansions
	2.7 Digit expansions of rational numbers
	2.8 Primitive roots
3 Integral solutions to the Pythagorean Equation
	3.1 Solutions
	3.2 Geometric method to find solutions
	3.3 Geometric method to find solutions: Non-Pythagorean examples
	3.4 Application: X4 + Y4 = Z4
4 What integers are areas of right triangles?
	4.1 Congruent numbers
	4.2 Small numbers
	4.3 Connection to cubic equations
5 What numbers are the edges of a right triangle?
	5.1 The theorem
	5.2 Gaussian integers
	5.3 The proof of Theorem 5.2
	5.4 Irreducible elements in mathbbZ[i]
	5.5 Proof of Theorem 5.1
6 Primes of the form 4k+1
	6.1 Euclid\'s theorem on the infinitude of primes
	6.2 Quadratic residues
	6.3 An application of the Law of Quadratic Reciprocity
7 Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol
	7.1 Gauss sums and Quadratic Reciprocity
	7.2 The Jacobi Symbol
8 Counting Pythagorean triples modulo an integer
	8.1 The Pythagorean Equation modulo a prime number p
	8.2 Solutions modulo n for a natural number n
9 How many lattice points are there on a circle or a sphere?
	9.1 The case of two squares
	9.2 More than two squares
	9.3 Integral points on arcs
10 What about geometry?
	10.1 Lattices in mathbbRn
	10.2 Minkowski\'s Theorem
	10.3 Sums of two squares
	10.4 Sums of four squares
	10.5 Sums of three squares
11 Another proof of the four squares theorem
	11.1 Quaternions
	11.2 Matrix representation
	11.3 Four squares
12 Quadratic forms and sums of squares
	12.1 Quadratic forms with integral coefficients
	12.2 Binary forms
	12.3 Ternary forms
	12.4 Three squares
13 How many Pythagorean triples are there?
	13.1 The asymptotic formula
	13.2 The computation of C2
14 How are rational points distributed, really?
	14.1 The real line
	14.2 The unit circle
A Background
	A.1  Sine, cosine, and exponentials
	A.2  The Binomial Theorem
	A.3  The Pigeon-Hole Principle
B Algebraic integers
C SageMath
	C.1  Basic operations
	C.2  Basic number theory
	C.3  Polynomial operations
References
		
Index