**Classroom Resource Materials**

Volume: 55;
2018;
153 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-4398-6
Product Code: CLRM/55**

List Price: $45.00

AMS Member Price: $33.75

MAA Member Price: $33.75

**Electronic ISBN: 978-1-4704-4846-2
Product Code: CLRM/55.E**

List Price: $45.00

AMS Member Price: $33.75

MAA Member Price: $33.75

#### Supplemental Materials

# Nuggets of Number Theory: A Visual Approach

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*Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

Nuggets of Number Theory will attract fans of visual thinking, number theory, and surprising connections. This book contains hundreds of visual explanations of results from elementary number theory. Figurate numbers and Pythagorean triples feature prominently, of course, but there are also proofs of Fermat's Little and Wilson's Theorems. Fibonacci and perfect numbers, Pell's equation, and continued fractions all find visual representation in this charming collection. It will be a rich source of visual inspiration for anyone teaching, or learning, number theory and will provide endless pleasure to those interested in looking at number theory with new eyes. Author Roger Nelsen is a long-time contributor of “Proofs Without Words” in the MAA's Mathematics Magazine and College Mathematics Journal. This is his twelfth book with MAA Press.

#### Readership

Undergraduate students, high school students, and teachers interested in number theory and visualization.

#### Reviews & Endorsements

The starting point for this book was the author's observation that many number theory texts contain few figures. He wrote an article for Math Horizons in 2008 that showed ways to use figures (the article was reprinted in the book "Biscuits of Number Theory"), and eventually expanded it to the present book. The book starts out with figurate numbers (squares, triangular numbers, etc.) that have an obvious picture, but quickly moves to other areas that we usually don't think of visually. This is not a complete introduction to number theory, but it does cover a good sampling, with something about congruences, Diophantine equations, irrational numbers, perfect numbers, and Fibonacci numbers.

-- Allen Stenger, MAA Reviews

#### Table of Contents

# Table of Contents

## Nuggets of Number Theory: A Visual Approach

- Cover i1
- Title page iv4
- Copyright v5
- Contents vi6
- Preface x10
- Chapter 1. Figurate Numbers 112
- Chapter 2. Congruence 2536
- Chapter 3. Diophantine Equations 3546
- Chapter 4. Pythagorean Triples 5566
- 4.1. Euclid’s formula 5667
- 4.2. Pythagorean triples and means of odd squares 5768
- 4.3. The carpets theorem 5869
- 4.4. Pythagorean triples and the factors of even squares 5970
- 4.5. Almost isosceles primitive Pythagorean triples 6172
- 4.6. A Pythagorean triple tree 6475
- 4.7. Primitive Pythagorean triples with square sides 6778
- 4.8. Pythagorean primes and triangular numbers 6778
- 4.9. Divisibility properties 6980
- 4.10. Pythagorean triangles 7081
- 4.11. Pythagorean runs 7384
- 4.12. Sums of two squares 7384
- 4.13. Pythagorean quadruples and Pythagorean boxes 7687
- 4.14. Exercises 7990

- Chapter 5. Irrational Numbers 8394
- 5.1. The irrationality of √2 8394
- 5.2. Rational approximations to √2: Pell equations 8899
- 5.3. Rational approximations to √2: Almost isosceles PPTs 89100
- 5.4. The irrationality of √3 and √5 90101
- 5.5. The irrationality of √𝑑 for non-square 𝑑 93104
- 5.6. The golden ratio and the golden rectangle 94105
- 5.7. The golden ratio and the regular pentagon 96107
- 5.8. Periodic continued fractions 98109
- 5.9. Exercises 101112

- Chapter 6. Fibonacci and Lucas Numbers 103114
- 6.1. The Fibonacci sequence in art and nature 104115
- 6.2. Fibonacci parallelograms, triangles, and trapezoids 106117
- 6.3. Fibonacci rectangles and squares 107118
- 6.4. Diagonal sums in Pascal’s triangle 113124
- 6.5. Lucas numbers 115126
- 6.6. The Pell equations 𝑥²-5𝑦²=±4 and Binet’s formula 117128
- 6.7. Exercises 121132

- Chapter 7. Perfect Numbers 123134
- 7.1. Euclid’s formula 123134
- 7.2. Even perfect numbers and geometric progressions 125136
- 7.3. Even perfect numbers and triangular numbers 126137
- 7.4. Even perfect numbers modulo 9 128139
- 7.5. Even perfect numbers end in 6 or 28 128139
- 7.6. Even perfect numbers modulo 7 130141
- 7.7. Even perfect numbers and sums of odd cubes 131142
- 7.8. Odd perfect numbers 131142
- 7.9. Exercises 133144

- Solutions to the Exercises 135146
- Bibliography 149160
- Index 151162
- Back Cover 154165